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(2) Contents. List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iii. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vi. 1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.1. Historical Perspective . . . . . . . . . . . . . . 1.2. Definintions and Notation . . . . . . . . . . . . 1.2.1. Graphs, multigraphs, and hypergraphs 1.2.2. Decompositions . . . . . . . . . . . . . 1.2.3. Rosa-type labelings . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 1 1 2 3 5 7. 2. Extensions of Rosa-Type Labelings . . . . . . . . . . 2.1. 1-Rotational Vertex Labelings . . . . . . . . . . . . 2.2. Rosa-Type Labelings for Multigraphs . . . . . . . . 2.2.1. λ-fold Rosa-type labelings . . . . . . . . . . 2.2.2. Results when λ = 2 . . . . . . . . . . . . . . 2.2.3. Results when λ > 2 . . . . . . . . . . . . . . 2.3. Vertex Labelings for Subgraphs of Circulant Graphs 2.3.1. Multipartite graphs . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 10 10 11 12 13 14 15 18. 3. Generating Infinite Classes of G-Decompositions 3.1. Generalization of α-labeling . . . . . . . . . . . . . 3.1.1. Ordered labeling for bipartite graphs . . . 3.1.2. Labelings for tripartite graphs . . . . . . . 3.2. Extensions into Multigraphs . . . . . . . . . . . . 3.2.1. Results when λ > 2 . . . . . . . . . . . . . 3.3. Extensions into Circulant Graphs . . . . . . . . . 3.3.1. Ordered labelings for multipartite graphs . 3.4. 1-Rotational Variants . . . . . . . . . . . . . . . . 3.4.1. 1-rotational simple graphs . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 21 21 22 23 31 34 36 38 38 39. 4. Alpha-Accommodation . . . . . . . . . . . 4.1. α-accommodating Labelings . . . . . . 4.2. α-accommodating Tripartite Labelings 4.2.1. Settling a cycle system . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 47 51 54 55. i. . . . .. . . . .. . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . ..
(3) ii. Contents 5. Decompositions of Hypergraphs . . . . . . . . . . . . . 5.1. Extension of Rosa-Type Labelings into Hypergraphs . 5.2. Extension of 1-Rotational Labelings into Hypergraphs 5.3. Settling a Hypergraph Spectrum . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 58 60 62 66. 6. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Labelings of Some Classes of Graphs and Multigraphs . 6.2. 1-Rotational ρ-tripartite Labelings . . . . . . . . . . . . 6.2.1. Some notation . . . . . . . . . . . . . . . . . . . 6.2.2. Path plus an edge . . . . . . . . . . . . . . . . . 6.3. α-accommodating Labelings . . . . . . . . . . . . . . . 6.3.1. A single odd cycle larger than C3 . . . . . . . . 6.3.2. An even cycle with an odd cycle larger than C3 6.3.3. An even cycle with a C3 . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 70 70 71 71 72 82 83 86 92. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99.
(4) List of Figures. 1.1 2.1 3.1. Graphical representation of the bridges of K¨onigsberg circa 1736. . . . . . . . . Various λ-fold Rosa-type labelings. . . . . . . . . . . . . . . . . . . . . . . . . . An α-labeling of an 8-cycle and the 3 starters obtained via Theorem 3.1 for a cyclic C8 -decomposition of K49 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 An ordered ρ-labeling of a 6-cycle and the 3 starters obtained by way of Theorem 3.2 for a cyclic C6 -decomposition of K37 . . . . . . . . . . . . . . . . . 3.3 A σ-tripartite labeling of a graph G with 8 edges. . . . . . . . . . . . . . . . . 3.4 The three copies of G (from Figure 3.3) obtained by way of Theorem 3.3 for a cyclic G-decomposition of K49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 A ρ-tripartite labeling of the Petersen graph P . . . . . . . . . . . . . . . . . . . 3.6 The three copies of P (from Figure 3.5) obtained by way of Theorem 3.5 for a cyclic P -decomposition of K91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 A 2-fold ρ+ -labeling of a multigraph G with 7 edges and the 3 starters obtained using Theorem 3.6 for a cyclic G-decomposition of 2K22 . . . . . . . . . . . . . . 3.8 A 3-fold ρ+ -labeling of a multigraph G with 9 edges and the 3 starters obtained using Theorem 3.7 for a cyclic G-decomposition of 3K19 . . . . . . . . . . . . . . 3.9 A 4-fold ρ+ -labeling of a multigraph G with 6 edges and the 3 starters obtained using Theorem 3.7 for a cyclic G-decomposition of 4K10 . . . . . . . . . . . . . . 3.10 Demonstrating a 1-rotational ρ-tripartite labeling of K3 + e. . . . . . . . . . . 3.11 Demonstrating a 1-rotational ρ-tripartite labeling of K3 + e choosing a different vertex tripartition than that found in Figure 3.10. . . . . . . . . . . . . . . . . 4.1 A K4 with a β-labeling that accommodates a K2,3 with an α-labeling. . . . . . 4.2 Two different σ-labelings of K4 ∪ K2,3 . . . . . . . . . . . . . . . . . . . . . . . 4.3 A ρ-tripartite labeling of C9 that α-accommodates sizes at least 2. . . . . . . . 4.4 An α-accommodating ρ-labeling of C9 ∪ C4 . . . . . . . . . . . . . . . . . . . . . 4.5 A σ-labeling of K2,3 ∪ C4 ∪ C4 ∪ K2,3 ∪ K4 . . . . . . . . . . . . . . . . . . . . . 4.6 An α-accommodating ρ-tripartite labeling of C9 ∪ C4 ∪ K2 ∪ C8 ∪ C4 . . . . . . 5.1 A 3-uniform hypergraph with 7 vertices and 3 edges. . . . . . . . . . . . . . . . 6.1 Examples of the path notations. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A 1-rotational ρ-tripartite labeling of P5 + P8 + P6 + (u, w). . . . . . . . . . . 6.3 A 1-rotational ρ-tripartite labeling of P1 + P5 + P1 + (u, w). . . . . . . . . . . 6.4 A 1-rotational ρ-tripartite labeling of (w, u) + P6 + P5 + P4 . . . . . . . . . . . 6.5 A 1-rotational ρ-tripartite labeling of P5 + P13 + P4 + (u, w). . . . . . . . . . . 6.6 A 1-rotational ρ-tripartite labeling of P1 + P10 + P4 + (u, w). . . . . . . . . . .. iii. . 1 . 13 . 22 . 23 . 24 . 26 . 28 . 31 . 34 . 35 . 36 . 45 . . . . . . . . . . . . . .. 45 48 49 51 53 54 55 59 72 74 76 76 77 79.
(5) iv. List of Figures 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16. A 1-rotational ρ-tripartite labeling of (w, u) + P5 + P10 + P4 . A 1-rotational ρ-tripartite labeling of P4 + P11 + P4 + (u, w). An α-accommodating ρ-tripartite labeling of C9 . . . . . . . . An α-accommodating ρ-tripartite labeling of C11 . . . . . . . An α-accommodating ρ-tripartite labeling of C9 ∪ C10 . . . . An α-accommodating ρ-tripartite labeling of C11 ∪ C10 . . . . An α-accommodating ρ-tripartite labeling of C11 ∪ C22 . . . . An α-accommodating ρ-tripartite labeling of C3 ∪ C10 . . . . An α-accommodating ρ-tripartite labeling of C3 ∪ C16 . . . . An α-accommodating ρ-tripartite labeling of C3 ∪ C10 ∪ C14 .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 79 81 83 85 86 89 91 93 95 97.
(6) Abstract. This dissertation describes new and improved methodology useful for obtaining decompositions of infinitely-many graphs from a single labeling of the vertices. In the beginning there is a brief overview of some historical events and results in combinatorics that motivate the research that follows. After establishing some terminology and notation including the various graphs considered, we turn to the Rosa-type labelings from whence the author’s research branches off. In the second chapter, definitions of the original Rosa-type labelings are presented and extended to simple graphs and multigraphs. Some of these extensions involve adding conditions that yield results for finding decompositions of infinite classes of graphs. Other extensions involve modifying the basics of the definition to get different types of decompositions. The third chapter is where we find many of the results that allow us to claim decompositions on infinite classes of graphs and multigraphs. The fourth chapter introduces an innovative method for combining certain Rosa-type labelings to yield results on the vertex-disjoint union of two or more simple graphs, allowing us to settle a decomposition result on a certain cycle system. The fifth chapter is where we find a demonstration on how to view Rosa-type labelings on hypergraphs and how this can extend to further decomposition results. The final chapter collects many applications of the results that appeared earlier in the dissertation.. v.
(7) Acknowledgments. As with any great endeavor, this dissertation could not have been completed without the help and kindness of many individuals along the way. Their support in getting me to where I am today should not go unrecognized. First, I would like to thank my supervisor Mariusz Meszka. His willingness to sponsor and guide me through this process was most gracious, and his constant suggestions and improvements on the work that appears herein immeasurably enhanced the final product. Next, I must also thank Saad El-Zanati whose friendship and collaboration I sincerely cherish. He not only got me started in doing research, but his influence on my life and career extends well beyond the bounds of academia. To my many coauthors (and future coauthors) too numerous to list here, I would like to extend my gratitude; I would, however, like to express thanks specifically to my frequent collaborators Charles Vanden Eynden and Daniel Roberts. Their fresh perspectives on frustrating problems are always helpful. Finally, I must extend my appreciation to my family for all of the support they have provided me along with the opportunities they have afforded me. I especially want to thank my wife Leslie for her enduring love and encouragement.. vi.
(8) Chapter 1. Introduction. 1.1. Historical Perspective The foundations of what we now call Graph Theory can be traced back to the early 1700s when Leonhard Euler showed that the then seven bridges found in the city of K¨onigsberg could not be traversed in a way such that each bridge is crossed exactly once [29]. We now would call this an Euler tour of the graph found in Figure 1.1. c a. d b. Figure 1.1: Graphical representation of the bridges of K¨onigsberg circa 1736.. A little over a century later in 1850, the Rev. Thomas Kirkman posed a scheduling problem [38]. Kirkman’s “schoolgirl problem” asked in essence whether fifteen students (schoolgirls) could be arranged in groups of three over the course of seven successive days such that no student was paired with another more than once. The popularity of this problem has led to such schedules, or designs, of groups of three where each member is placed in a group to be named Kirkman triple systems. In fact, a similar question was posited by Jakob Steiner [56], a contemporary of Kirkman. The Steiner triple systems similarly require groups, or blocks, of three where no two members of the larger set are in more than one group, but without the requirement of having to schedule every member all at the same time. Interestingly, the existence of these less-restrictive triple systems named for Steiner was settled by Kirkman [37] before he posed his schoolgirl problem. The recreational popularity of the latter 1.
(9) 1.2. Definintions and Notation. 2. problem seems to have kept Kirkman from getting his due credit (see [26] for a more detailed account). The Kirkman triple systems and the related Steiner systems can be viewed as a way of determining how to break apart or separate the many possible connections within a large group into smaller collections of those same connections. In the terminology of this dissertation, we call this a decomposition of the graph that represents those connections from the larger group into smaller graphs, often graphs with identical structure. To illustrate how seemingly simple descriptions can lead to graph decompositions that are difficult to find, consider the Oberwolfach problem posed in 1967 by Gerhard Ringel (cf. [41]), which can also be posed as a scheduling problem: You and an even number of your friends are meeting to have dinner at a banquet hall. The number of seats at the various tables, the sizes of which can differ, matches the number of attendees. Can a schedule of dinners be devised such that each person eventually sits next to everyone else exactly once? While special cases and certain exceptions of the Oberwolfach problem are known, the general solution remains unknown, still. The methods described in this dissertation to facilitate decompositions is through the use of assigning numbers, or labels, to the nodes in the graph. As mentioned above, determining whether or not a decomposition exists can be quite difficult. However, the new methodology introduced in this dissertation allows for decompositions of infinitely-many graphs, all of which can be obtained from a single labeling, i.e., assignment of integer values. Moreover, this dissertation outlines an innovative approach that breathes new life into known results to settle a certain type of cycle system, i.e., decompositions into certain types of cycles. In the last chapter, applications demonstrating the usefulness of these new methods can be found; however, an interested reader can find many more such applications by perusing the publication history of the author of this dissertation. Finally, the results presented here are heavily influenced by the seminal work done by Alex Rosa in the 1960s on the topic of graph labelings [51], and these results also expand on work done more recently by Saad El-Zanati and Charles Vanden Eynden [8, 28].. 1.2. Definintions and Notation Any notation or terminology used in this dissertation that is not outright defined within can be found in Graph Theory by Bondy and Murty [10]. We also turn the readers attention to the Handbook of Combinatorial Designs [24] where many of the.
(10) 1.2. Definintions and Notation. 3. preliminary results mentioned here can be found with specific attention to the area of design theory. If a and b are integers we denote {a, a + 1, . . . , b} by [a, b] (if a > b, then [a, b] = ∅). Let N denote the set of nonnegative integers and Zm the group of integers modulo m. For a finite set S and positive integer λ, we let λS denote the multiset that contains every element of S exactly λ times. For example 2 [4, 7] is the multiset {4, 4, 5, 5, 6, 6, 7, 7}. On the other hand, for S ⊆ Z and positive integer λ, we use λS to denote the set {λs : s ∈ S}, so 2[4, 7] = {8, 10, 12, 14}. 1.2.1. Graphs, multigraphs, and hypergraphs We define a graph G to be an ordered pair V (G), E(G) where V (G) is a set of vertices and E(G) is a set of 2-element subsets of V (G) called edges. The cardinality of the vertex set V (G) and the cardinality of the edge set E(G) are called the order and size of G, respectively. If we allow E(G) to be a multiset, then we call G a multigraph. For example, the structure seen in Figure 1.1 can be interpreted as the multigraph {a, b, c, d}, {a, b}, {a, b}, {a, c}, {a, c}, {a, d}, {b, d}, {c, d} . If we instead allow E(G) to contain subsets of V (G) with any number of vertices (not just 2), then we call G a hypergraph. If all edges of a hypergraph G are k-element subsets of V (G), then we say G is a k-uniform hypergraph, or has uniformity k. (Throughout this dissertation, we will use visual representations of graphs and multigraphs such as that seen in Figure 1.1 to imply the vertex and edge set.) Two graphs G and H are said to be isomorphic if there exists a one-to-one corre. spondence φ : V (G) → V (H) such that {u, v} ∈ E(G) if and only if φ(u), φ(v) ∈ E(H). Furthermore, if G and H are multigraphs, then the multplicity of edge {u, v} . in E(G) is equivalent to the multiplicity of edge φ(u), φ(v) in E(H). The definition of isomorphic hypergraphs extends naturally from that of isomorphic graphs. Certain types of graphs are used extensively throughout this dissertation. Consider the set V = {v1 , v2 , . . . , vn }. A graph with vertex set V and edge set {vi , vj } : vi , vj ∈. V is called a complete graph and is denoted KV . A graph with vertex set V and . edge set {vi , vi + 1} : 1 ≤ i ≤ n − 1 is called a path and is denoted PV . A graph . with vertex set V and edge set {vi , vi + 1} : 1 ≤ i ≤ n − 1 ∪ {vn , v1 } is called a cycle and is denoted CV . Note that any graph of order n that is isomorphic to KV , PV , or CV will be denoted as Kn , Pn , or Cn , respectively. A graph is said to be k-partite if its edge set can be partitioned into k subsets, called parts, such that no edge consists of vertices from the same part. We may refer to a k-partite graph as bipartite when k = 2 and tripartite when k = 3. A cycle of odd order is an example of a tripartite graph that is not bipartite. We note that.
(11) 1.2. Definintions and Notation. 4. popular convention is to refer to a graph as k-partite using only the least possible value of k for that graph. For example, cycles even order are usually referred to only as bipartite, not tripartite. For the purposes of some of the results in this dissertation, we intentionally do not make such a distinction. Consider the disjoint vertex sets V1 , V2 , . . . , Vk where |Vi | = vi for all i ∈ [1, k]. . S The graph with vertex set ki=1 Vi and edge set {ui , uj } : ui ∈ Vi , uj ∈ Vj , i 6= j is called a complete k-partite graph, or more generally a complete multipartite graph, and is denoted as KV1 ,V2 ,...,Vk . Note that any graph of that is isomorphic to KV1 ,V2 ,...,Vk may simply be denoted as a Kv1 ,v2 ,...,vk . Also, if v1 = v2 = · · · = vk = n, then we may use the notation Kk×n to denote this complete k-partite graph. A 1-factor of a graph H of even order is a subgraph of H with vertex set V (H) and such that the degree of each vertex in the subgraph is 1. A 1-factor is also called a perfect matching, as each vertex is paired up with another vertex to which it is adjacent in H. Often times, results/problems on odd-order complete graphs can be extended to/re-stated for larger complete graphs when a 1-factor is removed. (Such is the case for Steiner triple systems and the Oberwolfach problem, for example.) Consider a complete graph on 2k vertices and let I denote a 1-factor of that K2k . The graph commonly referred to as the “cocktail party” graph K2k − I, i.e. the graph with edge set E(K2k ) \ E(I), is isomorphic to the complete k-partite graph Kk×2 . Let G = (V, E) be a graph and λ a positive integer. The multigraph with vertex set V and edge multiset λE we denote by λ G. We call λKn a λ-fold complete multigraph of order n. (k). We use Kn to denote a complete k-uniform hypergraph of order n, i.e. the hypergraph of order n with an edge set that contains all k-element subsets of its vertex set. If e is an edge of graph G, then we use G − e to denote the subgraph with edge set E(G) \ {e}. If v is a vertex of a graph G, then we use G v to denote the induced subgraph on V (G) \ {v}. If G is a subgraph of KZn and i ∈ Zn , then we . use G + i to denote the graph with edge set {u + i, v + i} : {u, v} ∈ E(G) where addition is performed modulo n. We similarly define the notation G + i when G is a subgraph of Kn with V (Kn ) = Zn−1 ∪ {∞} to denote the graph with edge set . {u + i, v + i} : ∞ 6∈ {u, v} ∈ E(G) ∪ {∞, v + i} : {∞, v} ∈ E(G) with addition now performed modulo n − 1. Let S be a subset of Zn that does not contain 0. The graph with vertex set Zn . and edge set {i, j} : i − j ≡ s (mod n), s ∈ S is called a circulant graph and is denoted hSin . We note that hSin and hS 0 in can be isomorphic graphs even if S 6= S 0 . For example, if m > 1, then h[1, m]i2m+1 ∼ = h[m + 1, 2m]i2m+1 ∼ = K2m+1 . It follows.
(12) 1.2. Definintions and Notation. 5. then that any circulant graph on n vertices can be determined by a set S where max(S) ≤ bn/2c. 1.2.2. Decompositions Let G and H be graphs with G a subgraph of H and let Γ be a set of subgraphs of H. A decomposition of H is a set ∆ = {G1 , G2 , . . . , Gt } of edge-disjoint subgraphs S of H where ti=1 E(Gi ) = E(H). Each subgraph in ∆ is called a block. We call ∆ a Γ-decomposition of H if each block in ∆ is isomorphic to a graph in Γ. Furthermore, if Γ = {G}, then we call ∆ a G-decomposition of H. If there exists a G-decomposition of H, then we say G decomposes H and may write G | H. An automorphism of a G-decomposition of H is a one-to-one correspondence of the vertices of H to themselves such that each G-block is mapped to a G-block. A G-decomposition of H is also known as an (H, G)-design or, if H is a complete graph on n vertices, a G-design of order n. The connection of graph decompositions in graph theory to block designs in design theory is worth special note. A design, or block design, is an ordered pair (V, ∆) where V is a set (whose elements are called points) and ∆ is a set of subsets (called blocks) of V . There are many different types of designs, but for the purposes of brevity we look specifically at one such type and its connection to graph decompositions: A balanced incomplete block design, or BIBD, is a design (V, ∆) such that (i) each block is an m-element subset of V , (ii) each point appears in the same number of blocks, and (iii) any 2-element subset of V is contained in exactly λ blocks. Given such a BIBD (V, ∆) with λ = 1, we can thus represent ∆ as a Km -decomposition of KV . In fact, many results on the various block designs have graph decomposition analogues, including decompositions of hypergraphs and multigraphs. The concept of a graph decomposition extends naturally to multigraphs and hypergraphs. First, when G and H are hypergraphs with G a sub-hypergraph of H, a G-decomposition of H is a set ∆ = {G1 , G2 , . . . , Gt } of edge-disjoint sub-hypergraphs S (k) of H where ti=1 E(Gi ) = E(H). A G-decomposition of Kn can also be called a (k) Kn , G -design. Second, if H is a multigraph and G is a subgraph (or sub-multigraph) of H, then a G-decomposition of H is a set (or multiset) ∆ = {G1 , G2 , . . . , Gt } of subgraphs (or sub-multigraphs) of H where each Gi ∈ ∆ is isomorphic to G S and the multiplicity of an edge in ti=1 E(Gi ) matches its multiplicity in E(H). A G-decomposition of λKn is similarly called a λKn , G -design of index λ or a G-design of order n and index λ. Let V (Kn ) = Zn and let G be a subgraph of Kn . The length of an edge {i, j} ∈ E(G) is defined as min{|i − j|, n − |i − j|}. By clicking G, we mean applying the.
(13) 1.2. Definintions and Notation. 6. one-to-one mapping φ : V (G) → V (Kn ) defined as i 7→ i+1. Note that this results in a subgraph of Kn that is isomorphic to G. For convenience we will use φ(G) to denote this graph, which has vertex set {φ(v) : v ∈ V (G)} and edge set {φ(u), φ(v)} :. {u, v} ∈ E(G) . A G-decomposition ∆ of Kn is cyclic if clicking is an automorphism of ∆. That is to say φ(Gi ) ∈ ∆ if and only if Gi ∈ ∆. Furthermore, if |∆| = n, then we call it purely cyclic. Now, let V (Kn ) = Zn−1 ∪ ∞ and let G be a subgraph of Kn . The length of an edge {i, j} ∈ E(G) where {i, j} 63 ∞ is defined as min{|i − j|, n − 1 − |i − j|}; whereas, the an edge {i, ∞} ∈ E(G) is said to have length ∞. Similarly as above, clicking G still implies applying the isomorphism φ defined as i 7→ i + 1 for i ∈ Zn−1 , but we now incorporate the convention that φ(∞) = ∞. A G-decomposition ∆ of Kn is called 1-rotational if clicking (as defined here with ∞ 7→ ∞) is an automorphism of ∆. While the existence of Steiner triple systems—which we can now refer to as K3 -decompositions of complete graphs—has long been settled, determining the existence of a G-decomposition of an arbitrary graph, where G contains an edge sharing vertices with at least two other edges, has been shown to be NP-complete [35]. That is to say we expect not to find a computationally quick algorithm to solve the problem in general. That being said, results for certain classes of graphs can be found. As mentioned above, Kirkman settled the result for the existence of Steiner triple systems [37], and shortly after Kirkman’s schoolgirl problem appeared, Arthur Cayley [23] and Kirkman [39] both published solutions for that instance, i.e., for a set of 15 items/students. However, it was not until over a century later that the existence of Kirkman triple systems was settled [42, 50]. Triple systems can be viewed as a special case of the more general cycle systems. In short, cycle systems are decompositions of complete graphs where each of the blocks is a cycle (not just a triangle). The Oberwolfach problem (as described above) is an example of a cycle system. One can think of the Oberwolfach problem as finding a G-decomposition of Kn where G is a collection of disjoint cycles with a total size of n. To date, only four exceptions are known [52]; otherwise, it is conjectured that the necessary conditions for the Oberwolfach problem are sufficient. (We note that the Oberwolfach problem is also defined for an even number of dinner attendees with the removal of a 1-factor and direct the interested reader to [11] for more details on the known cases and exceptions.) Many of the applications in this dissertation pertain to cycles systems, i.e. decompositions of complete graphs into certain collections of cycles. However, it is generally our goal to generate these decompositions cyclically. Another classic cycle-related problem in graph theory is that of finding a Hamilton cycle as a subgraph. That is, given a graph of order n, does it have Cn as a.
(14) 1.2. Definintions and Notation. 7. subgraph? This problem, named for a contemporary of Kirkman, seems as simple as the known result on Euler tours. Alas, finding such a subgraph has been shown to be NP-complete [36]. (An interesting note: Kirkman actually posed this question a few years prior to Hamilton [40], but again the namesake to the latter.) However, hamiltonian decompositions, where every block is a Hamilton cycle, of complete graphs (and variations on complete graphs when n is even) have been known since the 1890s [43]. We can rest assured our search for decompositions of complete graphs will not be futile. In fact, given an arbitrary graph G, it has been shown [58] that G-decompositions of Kn exist for all sufficiently large n that satisfy certain base-level conditions that would be necessary for any graph decomposition: • the order of G does not exceed the order of Kn , i.e. n ≥ |V (G)|; • the size of Kn is divisible by the size of G, i.e. n2 ≡ 0 (mod |E(G)|); • the degree of each vertex in Kn can be written as as a linear combination of the degrees of vertices found in G, i.e. n − 1 ≡ 0 (mod d) where d = gcd{deg v : v ∈ V (G)}. The question, of course, still remains: For which values of n do we know a G-decomposition of Kn exists? To see progress on this question, we direct the reader to [14]. 1.2.3. Rosa-type labelings For any graph G, a one-to-one function f : V (G) → N is called a labeling (or a valuation) of G. In [51], Rosa introduced a hierarchy of labelings. Let G be a graph with n edges and no isolated vertices and let f be a labeling of G. Let f (V (G)) = {f (u) : u ∈ V (G)}. Define a function f¯ : E(G) → Z+ by f¯(e) = |f (u) − f (v)|, where e = {u, v} ∈ E(G). We will refer to f¯(e) as the label of e. Let f¯(E(G)) = {f¯(e) : e ∈ E(G)}. Consider the following conditions: (`1) f (V (G)) ⊆ [0, 2n], (`2) f (V (G)) ⊆ [0, n], (`3) f¯(E(G)) = {x1 , x2 , . . . , xn } where for each i ∈ [1, n] either xi = i or xi = 2n + 1 − i, (`4) f¯(E(G)) = [1, n]. If in addition G is bipartite with bipartition {A, B} of V (G), consider also (`5) for each {a, b} ∈ E(G) with a ∈ A and b ∈ B, we have f (a) < f (b),.
(15) 1.2. Definintions and Notation. 8. (`6) there exists an integer δ such that f (a) ≤ δ for all a ∈ A and f (b) > δ for all b ∈ B. Then a labeling satisfying the conditions (`1) and (`3) is called a ρ-labeling, (`1) and (`4) is called a σ-labeling, (`2) and (`4) is called a β-labeling. A β-labeling is necessarily a σ-labeling, which in turn is a ρ-labeling. Suppose G is bipartite. If a ρ-, σ-, or β-labeling of G satisfies condition (`5), then the labeling is ordered and is denoted by ρ+, σ +, or β +, respectively. If in addition (`6) is satisfied, the labeling is uniformly-ordered and is denoted by ρ++, σ ++, or β ++, respectively. A β-labeling is better known as a graceful labeling and a uniformly-ordered β-labeling is an α-labeling as introduced in [51]. It has been conjectured that every tree admits a β-labeling (more widely known as the “graceful tree conjecture”) [51]. Some partial results on this front include trees on up to 35 vertices [30], trees with diameter at most 5 [34], and caterpillars [51] and lobsters [6]. Labelings of the types above are called Rosa-type labelings because of Rosa’s original article [51] on the topic (see [27] for a survey of Rosa-type labelings). A dynamic survey on general graph labelings is maintained by Gallian [31]. Rosa-type labelings are critical to the study of cyclic graph decompositions as seen in the following result from [51]. Theorem 1.1. Let G be a graph with n edges. There exists a purely cyclic G-decomposition of K2n+1 if and only if G has a ρ-labeling. It is worth noting that results on cyclic, or even purely cyclic, decompositions predate Rosa’s work. Some of Kirkman’s original work [37] that settled what we now call Steiner triple systems made use of so cyclic permutations. Others, who seem likely to have been unaware of Kirkman’s work, approached the problem from a different perspective. For example, Heffter pursued cyclic Steiner triple systems by way of difference sets and published two famous problems [32, 33] in the 1890s. Heffter’s first and second difference problems would be solved by Peltesohn [49] forty years later, and another couple of decades later Skolem’s constructions for the two problems [53, 54] appeared, which would lead to the namesake “Skolem sequences”. However, in this dissertation we will be focusing on cyclic decompositions obtainable through graph labelings, so Theorem 1.1 is of utmost importance for what is to follow here..
(16) 1.2. Definintions and Notation. 9. While it is certainly the case that not every graph admits a ρ-labeling, many different classes of graphs have been shown to admit ρ-labelings [12]. In fact, it has been conjectured that all trees admit a ρ-labeling or better yet are graceful [51]. Given the complexity of such a claim, that problem may remain open indefinitely, but we can still limit our search for labelings of a graph based on the properties of that graph. Call a connected graph G Eulerian if every vertex of G has even degree. If a graph G with Eulerian components admits a σ-labeling, then we have the following well-known restriction on |E(G)|. Theorem 1.2. (Parity Condition in [51]) Let G be a graph with n edges comprised of Eulerian components. If G admits a σ-labeling, then n ≡ 0 or 3 (mod 4). A non-bipartite graph G is said to be almost-bipartite if G − e is bipartite for some e ∈ E(G). Note that if G is almost-bipartite with e = {ˆb, c}, then G is necessarily tripartite and V (G) can be partitioned into three sets A, B, and C = {c} such that ˆb ∈ B and e is the only edge joining an element of B to c. Let G be an almost-bipartite graph with n edges with vertex tripartition A, B, C as above. A labeling h of the vertices of G is called a γ-labeling of G if the following conditions hold. (g1) The function h is a ρ-labeling of G. (g2) If {a, v} is an edge of G with a ∈ A, then h(a) < h(v). (g3) We have h(c) − h(ˆb) = n. It was shown in [8] that if a graph G with n edges admits a γ-labeling, then there exists a cyclic G-decomposition of K2nx+1 for any positive integer x. Several classes of almost-bipartite graphs have been shown to have γ-labelings (see [27]). In particular, it was shown in [8] that every cycle of odd length at least 5 admits a γ-labeling..
(17) Chapter 2. Extensions of Rosa-Type Labelings. Alex Rosa established the initial hierarchy of graph labelings—ρ, σ, β, and α—seen in Chapter 1. Extensions soon followed, including the ordered labelings from Saad El-Zanati, Charles Vanden Eynden, et al. Here we extend their ideas even further to accomplish more decompositions. It is worth noting that all of the definitions found in this chapter are extended further in Chapter 3. In the first section of this chapter, we connect Rosa-type labelings to a labeling that induces a 1-rotational decomposition. This approach, while implied in previous work, is codified here to align with results seen elsewhere in this dissertation as well as in the literature. The second section sees an extension of Rosa-type labelings to multigraphs. As a side note, many of the definitions in this section first appeared in [18] that this author completed with El-Zanati, Vanden Eynden, and others. The final section of this chapter points to a natural connection of Rosa-Type labelings to circulant graphs. The definitions and corresponding results found in this and the following chapter have already shown to be most useful in the on-going hunt for decompositions (i.e., settling graph spectrum problems); see the “small designs” in [22] or the designs used to prove some of the lemmas in [3] for recent examples of applications of the concepts defined in this chapter.. 2.1. 1-Rotational Vertex Labelings The Rosa-type labelings described in Section 1.2.3 all yield decompositions of complete graphs of orders that are 1 more than a multiple of twice the number of edges of a graph G. However, it is often desirable to find G-decompositions of complete graphs of other orders. Namely, complete graphs of orders that are a multiple of
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(19) 2 ·
(20) E(G)
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(23) set of integers modulo 2 ·
(24) E(G)
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(26) 2.2. Rosa-Type Labelings for Multigraphs. 11. graph, we quickly find that we can no longer obtain a decomposition cyclically, so an alternative approach is to decompose the complete graph 1-rotationally, i.e., through the use of a fixed point, commonly denoted with vertex label ∞, around which the rest of the vertices still act as in the case of cyclic decompositions. Through the use of edge lengths and clicking, a 1-rotational decomposition can be viewed through the lens of Rosa-type labelings. Let G be a graph with n edges, no isolated vertices, and a vertex w of degree 1. A 1-rotational ρ-labeling of G is a one-to-one function f : V (G) → [0, 2n − 2] ∪ {∞} where f (w) = ∞ and such that f is a ρ-labeling of G w. This novel description of a Rosa-type labeling allows for extensions to and connection with other known results. We start first with the following natural analogue of Theorem 1.1. Theorem 2.1. Let G be a graph with n edges. There exists a 1-rotational G-decomposition of K2n if and only if G admits a 1-rotational ρ-labeling. Proof. Let w ∈ V (G) have degree 1. Sufficiency follows directly from Theorem 1.1 and that a 1-rotational ρ-labeling with w 7→ ∞ induces a ρ-labeling on G w. To show necessity, consider a 1-rotational G-decomposition of K2n , say ∆. We note that K2n with vertex set Z2n−1 ∪ ∞ has exactly 2n − 1 edges of each length ` ∈ [1, n − 1]. Furthermore, edge length is preserved under clicking. Thus, any G0 ∈ ∆ must consist of no more than one edge of each length ` ∈ [1, n − 1]. Since G0 must have n edges to be isomorphic to G, it must have exactly one edge of each length in K2n and one edge incident with ∞. Therefore, the vertices in G0 induce a 1-rotational ρ-labeling of G.. 2.2. Rosa-Type Labelings for Multigraphs A ρ-labeling of a graph G with n edges is essentially an embedding of G in K2n+1 that induces one edge in G of length i for each i ∈ [1, n]. Clicking this embedding of G in K2n+1 produces a purely cyclic (K2n+1 , G)-design. If G is bipartite and the ordered property holds in the embedding, then we can obtain a cyclic (K2nx+1 , G)-design for every positive integer x (see Theorem 3.2 in the next chapter or also [28]). These concepts extend naturally to embeddings in λK2n/λ+1 of a multigraph G with n edges. In such an embedding, G would need to contain the appropriate number of edges of each length. Clicking G would in turn yield a cyclic (λK2n/λ+1 , G)-design. As seen in Chapter 3, if the ordered property holds in the embedding, then a cyclic (λK2nx/λ+1 , G)-design can be obtained for every positive integer x. While every graph G with n edges and no isolated vertices can be embedded in K2n+1 , additional restrictions are needed for embeddings of G in λK2n/λ+1 . For example, the order of G.
(27) 12. 2.2. Rosa-Type Labelings for Multigraphs. cannot exceed 2n/λ + 1, which necessarily excludes simple forests consisting of more than one tree when λ ≥ 2. 2.2.1. λ-fold Rosa-type labelings Let n, k, and λ be positive integers such that n is either λk or λk + λ/2. Let G be a multigraph of size n, order at most 2n/λ + 1, and edge multiplicity at most λ. Let f be a labeling of G. For e = {u, v} ∈ E(G), let f¯(e) = |f (u) − f (v)| as before, but let f¯∗ (e) = min{f¯(e), 2n/λ + 1 − f¯(e)}. Consider the following conditions: (`0 1) f (V (G)) ⊆ [0, 2n/λ], (`0 2) either n = λk and f¯∗ (E(G)) = λ [1, k] or n = λk + λ/2 and f¯∗ (E(G)) = λ [1, k] ∪ (`0 3) either n = λk and f¯(E(G)) = λ [1, k] or n = λk + λ/2 and f¯(E(G)) = λ [1, k] ∪. λ/2. λ/2. {k + 1},. {k + 1}.. If in addition G is bipartite with vertex bipartition {A, B}, consider also the following conditions: (`0 4) for each {a, b} ∈ E(G) with a ∈ A and b ∈ B, we have f (a) < f (b); (`0 5) there exists an integer δ such that f (a) ≤ δ for all a ∈ A and f (b) > δ for all b ∈ B. Then a labeling satisfying the conditions: (`0 1) and (`0 2) is called a λ-fold ρ-labeling; (`0 1) and (`0 3) is called a λ-fold σ-labeling. As with Rosa’s original labelings, we similarly have that a λ-fold σ-labeling is necessarily a λ-fold ρ-labeling. If G is bipartite and a λ-fold ρ- or σ-labeling f of G satisfies condition (`0 4), then the labeling is ordered and is denoted by ρ+ or σ +, respectively. If in addition (`0 5) is satisfied, the labeling is uniformly-ordered and is denoted by ρ++ or σ ++, respectively. Figure 2.1 shows examples of multigraphs with various λ-fold labelings. In certain cases, the concept of a λ-fold σ-labeling coincides with that of a complete λ-equitable labeling (see [5] and [9]). The two concepts are the same when λ = 2 and the size of the graph is even. We next show that a graceful labeling of a graph G with n edges is necessarily a 2-fold ρ-labeling of G. However, there are graphs with size n and order at most n + 1 that admit a 2-fold ρ+-labeling, but do not admit a graceful labeling. For example,.
(28) 13. 2.2. Rosa-Type Labelings for Multigraphs 0. 1. 5. 3. 0. 2. 6. 6. (a). 1. 2. 0. 5. 3. 8. (b). 1. 3. 0. 7. 4. 4. (c). 1. 3. 2. 5. (d). Figure 2.1: (a) A 2-fold σ-labeling of C6 , (b) a 2-fold ρ++-labeling of C6 , (c) a 2-fold ρ++-labeling of a multigraph with 9 edges, and (d) a 3-fold ρ+-labeling of that same multigraph.. C6 fails the Parity Condition (see Theorem 1.2) and as such does not admit a graceful labeling, but C6 does admit a 2-fold ρ++-labeling as seen in Figure 2.1(b). Lemma 2.2. If G is a graph and f is a β-labeling of G, then f is a 2-fold ρ-labeling of G. Proof. Let G be a graph with n edges and let f be a β-labeling of G. By the definition of a β-labeling, f¯(E(G)) = [1, n]. Now by the definition of a 2-fold ρ-labeling, if e ∈ E(G), then f¯∗ (e) = min{f¯(e), n + 1 − f¯(e)}. Thus, f¯∗ (E(G)) = 2 [1, n/2] if n is even, and f¯∗ (E(G)) = 2 [1, (n − 1)/2] ∪ {(n + 1)/2} if n is odd. Therefore, f is a 2-fold ρ-labeling of G. Since the 2-fold ρ-labeling of G in the previous result is identical to the (1-fold) β-labeling of G, We necessarily have the following result on ordered labelings. Corollary 2.3. If G is a bipartite graph and f is a (uniformly) ordered β-labeling of G, then f is a (uniformly) ordered 2-fold ρ-labeling of G. 2.2.2. Results when λ = 2 To simplify the proofs of our λ-fold labelings results, we focus first on the case λ = 2 and show that the following extension of Theorem 1.1 must hold. Theorem 2.4. Let G be a subgraph of 2Kn+1 such that |E(G)| = n. There exists a purely cyclic G-decomposition of 2Kn+1 if and only if G admits a 2-fold ρ-labeling. Proof. Let G be a subgraph of 2Kn+1 such that |E(G)| = n. We separate the proof into two cases depending on whether n is even or odd. Case 1: Suppose n is even. Let n = 2k and let G admit a 2-fold ρ-labeling. Then for each length ` ∈ [1, k], there exist two edges of length ` in G. Denote these edges by e0` and e00` . Let G0 be the subgraph of G with E(G0 ) = {e0` : ` ∈ [1, k]}. Similarly, let G00 be the subgraph with E(G00 ) = {e00` : ` ∈ [1, k]}. Then G0 and G00 are edge disjoint, and neither of them.
(29) 2.2. Rosa-Type Labelings for Multigraphs. 14. contains double-edges. Moreover, the 2-fold ρ-labeling of G induces simultaneously a ρ-labeling of G0 and a ρ-labeling of G00 . Thus ∆G0 = {G0 + i : i ∈ [0, 2k]} is a purely cyclic (K2k+1 , G0 )-design. Similarly, ∆G00 = {G00 + i : i ∈ [0, 2k]} is a purely cyclic (K2k+1 , G00 )-design. Since G = G0 ∪ G00 and 2k + 1 = n + 1, the set ∆G = {G + i : i ∈ [0, 2k]} is a purely cyclic (2Kn+1 , G0 )-design. Conversely, any G-block in a purely cyclic (2Kn+1 , G)-design, induces a 2-fold ρ-labeling of G. Case 2: Suppose n is odd. Let n = 2k + 1 and let G admit a 2-fold ρ-labeling. As in the previous case, let e0` and e00` denote the two edges in G of length ` for each ` ∈ [1, k]. Also, let ek+1 be the edge of length k + 1. Let G0 be the subgraph of G with E(G0 ) = {e0` : ` ∈ [1, k]}, let G00 be the subgraph with E(G00 ) = {e00` : ` ∈ [1, k]}, and let Gk+1 be the subgraph with E(Gk+1 ) = {ek+1 }. Let I be the 1-factor in K2k+2 induced by the edges of length k + 1. Since G0 does not contain an edge of length k + 1, the set ∆G0 = {G0 + i : i ∈ [0, 2k + 1]} is a purely cyclic (K2k+2 − I, G0 )-design. Similarly, the set ∆G00 = {G00 + i : i ∈ [0, 2k + 1]} is a purely cyclic (K2k+2 − I, G00 )-design. Also, the set ∆Gk+1 = {Gk+1 + i : i ∈ [0, 2k + 1]} is the multigraph obtained by replacing each edge in I with a double edge. Since G = G0 ∪ G00 ∪ Gk+1 and 2k + 2 = n + 1, the set ∆G = {G + i : i ∈ [0, 2k + 1]} is a purely cyclic (2Kn+1 , G)-design. Conversely, any G-block in a purely cyclic (2Kn+1 , G)-design, induces a 2-fold ρ-labeling of G. 2.2.3. Results when λ > 2 The following is an extension of Theorem 2.4 to λ-fold. The proof is similar to that of Theorem 2.4. Theorem 2.5. Let n and λ be positive integers such that n ≡ 0 or λ/2 (mod λ). Let G be a subgraph of λK2n/λ+1 such that |E(G)| = n. There exists a purely cyclic G-decomposition of λK2n/λ+1 if and only if G admits a λ-fold ρ-labeling. Proof. Let G be a subgraph of λK2n/λ+1 such that |E(G)| = n. We separate the proof into two cases depending on the values of λ and n. Case 1: Suppose n ≡ 0 (mod λ). Let n = λk and let G admit a λ-fold ρ-labeling. Then for each length ` ∈ [1, k], there exist λ edges of length ` in G. Denote these edges by e`,1 , e`,2 , . . . , e`,λ . For each i ∈ [1, λ] let Gi be the subgraph of G with E(Gi ) = {e`,i : ` ∈ [1, k]}. Note that Gi and Gj are edge disjoint for i 6= j, and each Gi is a simple graph. Moreover, the λ-fold ρ-labeling of G induces a (1-fold) ρ-labeling on each Gi , i ∈ [1, λ]. Thus ∆Gi = {Gi + j : j ∈ [0, 2k]} is a purely cyclic (K2k+1 , Gi )-design for all i ∈ [1, λ]. S Since G = λi=1 Gi and 2k + 1 = 2n/λ + 1, the set ∆G = {G + j : j ∈ [0, 2k]} =.
(30) 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. 15. {Gi + j : i ∈ [1, λ], j ∈ [0, 2k]} is a purely cyclic (λK2n/λ+1 , G)-design. Conversely, any G-block in a purely cyclic (λK2n/λ+1 , G)-design, induces a λ-fold ρ-labeling of G. Case 2: Suppose λ is even and n ≡ λ/2 (mod λ). Let λ = 2t for some positive integer t, let n = 2tk + t, and let G admit a λ-fold ρ-labeling. Hence, there are 2t edges in G of each length ` ∈ [1, k] and t edges of length k + 1. For each ` ∈ [1, k], partition the edges of length ` in to pairs E`,1 , E`,2 , . . . , E`,t . That is, |E`,i | = 2 and E`,i ∩ E`,j = ∅ for all i 6= j. Furthermore, denote the edges of length k + 1 by e1 , e2 , . . . , et . Now, for each i ∈ [1, t] let Gi be the S subgraph (or sub-multigraph) of G with E(Gi ) = {ei } ∪ k`=1 E`,i . Note that Gi and Gj are edge disjoint for i 6= j, and each Gi has edge multiplicity at most 2. Moreover, the λ-fold ρ-labeling of G induces a 2-fold ρ-labeling on each Gi , i ∈ [1, t]. Thus ∆Gi = {Gi + j : j ∈ [0, 2k + 1]} is a purely cyclic (2K2k+2 , Gi )-design for all i ∈ [1, t]. S Since G = ti=1 Gi and 2k+2 = 2n/λ+1, the set ∆G = {G+j : j ∈ [0, 2k+1]} = {Gi + j : i ∈ [1, t], j ∈ [0, 2k + 1]} is a purely cyclic (λK2n/λ+1 , G)-design. Conversely, any G-block in a purely cyclic (λK2n/λ+1 , G)-design, induces a λ-fold ρ-labeling of G.. 2.3. Vertex Labelings for Subgraphs of Circulant Graphs The concept of edge length in KZn provides us with a natural connection between Rosa-type labelings and circulant graphs. If a set of integers S has max(S) < bn/2c, then the circulant graph hSin can be viewed as the graph with edges set comprised of all edges in KZn that have length ` ∈ S. We can thus provide the following extension of Rosa’s original labelings. Let G be a graph with n edges and no isolated vertices, let S be a set of positive integers with |S| = n, and let t be an integer with t > 2 max(S). Let f : V (G) → N be a vertex labeling of G with f¯: E(G) → Z+ the induced edge labeling function. Consider the following conditions: (c1) f (V (G)) ⊆ [0, t − 1], (c2) f (V (G)) ⊆ [0, max(S)], (c3) f¯(E(G)) = {x1 , x2 , . . . , xn } where for each i ∈ S either xi = i or xi = t − i, (c4) f¯(E(G)) = S. If in addition G is bipartite with bipartition {A, B} of V (G), consider also (c5) for each {a, b} ∈ E(G) with a ∈ A and b ∈ B, we have f (a) < f (b), (c6) there exists an integer δ such that f (a) ≤ δ for all a ∈ A and f (b) > δ for all b ∈ B..
(31) 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. 16. Then a labeling satisfying the conditions (c1) and (c3) is called a ρS -labeling, (c1) and (c4) is called a σS -labeling, (c2) and (c4) is called a βS -labeling. As with Rosa’s original hierarchy, a βS -labeling is necessarily a σS -labeling, which in turn is a ρS -labeling. Suppose G is bipartite. If a ρS -, σS -, or βS -labeling of G satisfies + + condition (c5), then the labeling is ordered and is denoted by ρ+ S , σS , or βS , respectively. If in addition (c6) is satisfied, the labeling is uniformly-ordered and is denoted ++ ++ by ρ++ S , σS , or βS , respectively. We also refer to a uniformly-ordered βS -labeling as an αS -labeling. Collectively, we may refer to these as circulant Rosa-type lableings. For particular sets S, the concept of a ρS -, σS -, or βS -labeling may be known by another name. For the special case where S = [k, k + n − 1], then all circulant Rosa-type labelings defined here match definitions this author and others gave in [20]. Also for S = [k, k + n − 1], what we call here a βS -labeling has been referred to as a k-graceful labeling by Slater [55] and by Mah´eo and Thuillier [44]. Many of the previous theorems on Rosa-type labelings extend to circulant graphs. Consider, for example, the following extension of Theorem 1.1, which follows directly from the definitions of circulant and ρS -labeling. Theorem 2.6. Let G be a graph with n edges and let S be an n-element subset of the positive integers with 2 max(S) < t as in the definition for ρS -labeling. There exists a purely cyclic G-decomposition of hSit if and only if G admits a ρS -labeling. Of special note are the circulant Rosa-type labelings that satisfy condition (c4). Since the set of edge lengths on a graph with such a σS -labeling is identical to its set of edge labels, we can conclude even more than that which appears in Theorem 2.6. Theorem 2.7. Let G be a graph with n edges and let S be an n-element subset of the positive integers. If G admits a σS -labeling f , then there exists a purely cyclic . G-decomposition of hSit for any integer t > max f (V (G)), 2 max(S) . Proof. By the definition of a σS -labeling, there exists an integer t0 > 2 max(S) such f (V (G)) ⊆ [0, t0 − 1]. Also, for each ` ∈ S there exists a unique edge in G with edge S label `, which is also its length in KZt for any t > t0 . Thus, t−1 i=0 E(G + i) = E hSit , and {Gi : i ∈ Zt } is a purely cyclic G-decomposition of hSit . Theorem 2.7 gives us a slightly stronger result than Theorem 2.6 in that we can obtain decompositions of an infinite class of circulant graphs. However, we do need to heed the following extensions of the Parity Condition (Theorem 1.2) depending.
(32) 17. 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. on which circulant Rosa-type labeling we are wanting to employ and which circulant graph we are wanting to decompose. Theorem 2.8. Let G be a graph with n edges comprised of Eulerian components and let S be an n-element subset of the positive integers. If G admits a σS -labeling, then P `∈S ` is even. Proof. Let f be a σS -labeling of G with induced edge labeling function f¯. Hence, f¯(E(G)) = S. Now, for each v ∈ V (G), let d+ v denote the number of edges {u, v} ∈ E(G) such that f¯ {u, v} = f (v) − f (u); similarly, let d− v denote the number of edges − ¯ {u, v} ∈ E(G) such that f {u, v} = f (u) − f (v). Note that d+ v + dv = deg v, which by the hypothesis is even. It follows that X. `=. `∈S. X. |f (u) − f (v)| =. {u,v}∈E(G). X. X + − d+ (dv − d− v · f (v) − dv · f (v) = v ) · f (v).. v∈V (G). v∈V (G). − + − Since d+ v + dv is even for all v ∈ V (G), so is dv − dv , and the result follows.. Theorem 2.9. Let G be a graph with n edges comprised of Eulerian components and let S be an n-element subset of the positive integers. If there exists a purely cyclic P G-decomposition of hSit for some even integer t > 2 max(S), then `∈S ` is even. Proof. Let hSit admit a purely cyclic G-decomposition. It follows from Theorem 2.6 that there exists a ρS -labeling f with induced edge labeling function f¯ such that f¯(E(G)) = {x1 , x2 , . . . , xn } where for each i ∈ S either xi = i or xi = t − i. Let E1 denote the set of edges {u, v} in G with |f (u) − f (v)| ∈ S and let E2 = E(G) \ E1 . Now, for each v ∈ V (G), let d+ v,1 denote the number of edges {u, v} ∈ E1 such that f¯ {u, v} = f (v) − f (u) ∈ S and let d+ v,2 denote the number of edges {u, v} ∈ E2 ¯ such that t − f {u, v} = t − f (u) − f (v) ∈ S; similarly, let d− v denote the number of edges {u, v} ∈ E1 such that f¯ {u, v} = f (u) − f (v) ∈ S and let d− v,2 denote the ¯ number of edges {u, v} ∈ E2 such that t − f {u, v} = t − f (v) − f (u) ∈ S. Note + − − that d+ v,1 + dv,2 + dv,1 + dv,2 = deg v, which by the hypothesis is even. It follows that X `∈S. `=. X. |f (u) − f (v)| +. {u,v}∈E1. =. X. X. t − |f (u) − f (v)|. {u,v}∈E2. X − + − · f (v) d+ · f (v) − d · f (v) + |E | · t + d · f (v) − d 2 v,1 v,2 v,2 v,1 v∈V (G). v∈V (G). = |E2 | · t +. X. + − − (d+ v,1 + dv,2 − dv,1 − dv,2 ) · f (v).. v∈V (G).
(33) 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. 18. + − − + + − − Since d+ v,1 + dv,2 + dv,1 + dv,2 is even for all v ∈ V (G), so is dv,1 + dv,2 − dv,1 − dv,2 , and the result follows.. We restate the above theorems in the following corollary to highlight the implied conditions on the set of edge lengths in S when searching for circulant Rosa-type labelings. Corollary 2.10. Let G be a graph with n edges comprised of Eulerian components and let S be an n-element subset of the positive integers. There exists a purely cyclic G-decomposition of hSit for some even integer t > 2 max(S), only if the number of odd elements in S is even. Furthermore, there exists a σS -labeling of G only if the number of odd elements in S is even. 2.3.1. Multipartite graphs As mentioned above, certain sets of edge lengths from the complete graph KZt lead to circulant graphs of particular interest. Consider when t is a composite number, say t = c · d. Then, the circulant graph with vertex set Zt generated by the set S = {1, 2, . . . , bt/2c} \ {i · c : i ∈ Z}, i.e. hSit , is a complete multipartite graph Kc×d . To further illustrate this, consider that edge lengths that are a multiple of c in KZt , where t = cd, must have end-vertices in the same congruence class modulo c even if the length of an edge {u, v} is computed as t − |u − v|. Hence, removing the edges with lengths that are multiple of c from KZcd results in a Kc×d with vertex partition . {v ∈ Zcd : v ≡ i (mod c)} : i ∈ Zc . We can now view cyclic decompositions of complete multipartite graphs through the lens of Rosa-type labelings, such as the next result that follows directly from Theorem 2.6. Theorem 2.11. Let c and d be positive integers, let S = {1, 2, . . . , bcd/2c} \ {i · c : i ∈ Z}, and let G be a graph with |S| edges. If G admits a ρS -labeling with cd satisfying the conditions on t as seen in the definition for a ρS -labeling, then there exists a cyclic G-decomposition of Kc×d . We also note some necessary restrictions on c an d required to apply the definition of the circulant Rosa-type labelings and their results. Lemma 2.12. Let c and d be positive integers with c even, let S = {1, 2, . . . , cd/2} \ {i · c : i ∈ Z}, and let G be a graph with |S| edges. If G admits a ρS -labeling with cd satisfying the conditions on t as seen in the definition for a ρS -labeling, then d must be even..
(34) 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. 19. Proof. Recall from the definition of the circulant Rosa-type labelings that t = cd > 2 max(S). Suppose for the sake of contradiction that d is odd. Then cd/2 is not a multiple of c and is thus included in set S. In fact, max(S) = cd/2. Thus we arrive at the contradiction cd > 2 max(S) = cd. Hence, d must be even. The result from the above lemma can also be viewed as check on the formula for the size of the graph with this particular circulant Rosa-type labeling seen in the following. Lemma 2.13. Let c and d be positive integers, let S = {1, 2, . . . , bcd/2c} \ {i · c : i ∈ Z}, and let G be a graph with |S| edges. If G admits a ρS -labeling with cd satisfying the conditions on t as seen in the definition for a ρS -labeling, then |S| = (c − 1) · d/2. Proof. Let bcd/2c = m. Since the set of edge lengths S is comprised of the consecutive positive integers no more than m taking out the multiples of c, we can partition S into the following subsets: {1, 2, . . . , c − 1}, {c + 1, c + 2, . . . , 2c − 1},. . . , b(d − 3)/2c ·. c + 1, b(d − 3)/2c · c + 2, . . . , b(d − 1)/2c · c − 1 , and b(d − 1)/2c · c + 1, b(d − 1)/2c · . c + 2, . . . , bcd/2c \ b(d + 1)/2c · c . Clearly, the first b(d − 1)/2c partite sets contain c − 1 elements each. The question remains how many elements are in last partite set and whether or not bcd/2c is the multiple of c represented by b(d + 1)/2c · c. By Lemma 2.12, we cannot have c even and d odd. In the case where c and d are both odd, we have bcd/2c = (cd − 1)/2, which is not a multiple of c, and the last partite set consists of elements {(cd − c + 2)/2, (cd − c + 4)/2, . . . , (cd − 1)/2}, which contains (c − 1)/2 elements. In the case where d is even, we have bcd/2c = c · d/2, which is a multiple of c, and the last partite set consists of elements {(cd − 2c + 2)/2, (cd − 2c + 4)/2, . . . , (cd − 2)/2}, which contains c − 1 elements. Thus, counting the number of elements per partite set yields the formula (c − 1) · (d − 1)/2 + (c − 1)/2 for d odd, |S| = (c − 1) · (d − 2)/2 + (c − 1) for d even. In either case, the formula simplifies to |S| = (c − 1) · d/2. Again, for the particular set S and vertex labelings that yields cyclic decompositions of complete multipartite graphs, the concept of a ρS -, σS -, or βS -labeling may be known by another name. Particularly, when d = 1, i.e. c = t, then the circulant graph hSit ∼ = Kt , and all of the circulant Rosa-type labelings match their original (simple) Rosa-type labeling counterpart. When d ≤ 2 and we have a graph that admits a σS without using vertex label 2 max(S) + 1, then that σS -labeling (or βS - or αS -labeling) again would satisfy the conditions (`1), (`2), and/or (`4) for a σ-labeling (or β- or.
(35) 2.3. Vertex Labelings for Subgraphs of Circulant Graphs. 20. α-labeling). It is also of note that when c = 1 the complete multipartite graph K1×d is just an empty graph on t vertices; the results in this case, however trivial, still hold. Furthermore, what we call here βS -labelings and αS -labelings (where S and consequently c and d are defined as in Theorem 2.11) coincide with what was referred to as d-graceful labelings and d-graceful α-labelings, respectively, by Pasotti [48]. We further note that the cyclic decompositions of complete multipartite graphs in this section are in similar to “purely cyclic” decompositions in that they a single G-block embedded in Kc×d with V (Kc×d ) = Zcd will require clicking G a total of cd times before it will come back on itself. This differs from an alternative, popular description of “cyclic” decompositions of Kc×d that requires V (Kc×d ) = Zc × Zd and subsequently an alternative definition of “clicking” where we add 1 to either the first or the second coordinate (but not both) of the ordered pair for each vertex. One major advantage of the cyclic decompositions of complete multipartite graphs as explored in this section over this alternative is the need to find only a fraction of the base blocks required to generate the cyclic decomposition. Indeed, the simple mapping φ : Zcd → Zc × Zd defined by φ(x) 7→ (q, r) where q and r are the unique integers (in Zc and Zd , respectively) that satisfy x = qd + r can be used to map any of the cyclic G-decompositions found using the methods described in this section to the alternative. However, it should be noted that this mapping of G-decompositions is not invertible..
(36) Chapter 3. Generating Infinite Classes of G-Decompositions. All of the results on Rosa-type labelings mentioned thus far pertain to finding a single decomposition from a single labeling. In this chapter, we showcase labelings that yield infinite classes of complete graphs, starting with the prior results known for the labelings given in Chapter 1 and continuing the theme of extending them to new types of graphs. The first section of this chapter starts with Rosa’s original α-labeling [51] and continues with the ordered labelings on bipartite graphs [28]. This author’s published contributions to the techniques presented in this chapter begin with the extension to tripartite graphs in the latter part of the first section (also see [16]) and the extensions of ordered labelings to multigraphs in the following section (also see [18]). The remainder of the chapter showcases ongoing research on extensions to circulant graphs and research recently submitted for publication (see [15]).. 3.1. Generalization of α-labeling First, we see the result by Rosa in [51] on an infinite class of complete graphs that can be decomposed through the use of an α-labeling and show a proof for this result. Theorem 3.1. Let G be a bipartite graph with n edges. If G admits an α-labeling, then there exists a cyclic G-decomposition of K2nx+1 for all positive integers x. Proof. Let h be an α-labeling of G with vertex bipartition {A, B} such that h(A) < h(B). If x = 1, the result follows from Theorem 1.1, we now assume x > 1. For 1 ≤ j ≤ x define hj : V (G) → [0, nx] by. hj (v) =. h(v). v ∈ A,. h(v) + (j − 1)n v ∈ B. 21.
(37) 22. 3.1. Generalization of α-labeling. 0. 1. 2. 3. 0. 1. 2. 3. 0. 8. 7. 5. 4. 8. 7. 5. 4. 16 15 13 12. (a) α-labeling of C8. 1. 2. 3. 0. 1. 2. 3. 24 23 21 20. (b) Three starters for a cyclic (K49 , C8 )-design. Figure 3.1: An α-labeling of an 8-cycle and the 3 starters obtained via Theorem 3.1 for a cyclic C8 -decomposition of K49 .. We define a multigraph H with vertex set contained in [0, nx] and with the nx edges {hj (u), hj (v)}, one for each edge {u, v} of G and 1 ≤ j ≤ x. For 1 ≤ i ≤ nx, we need to show that H has an edge with label i = qn + r,. 1 ≤ r ≤ n,. 0 ≤ q < x.. By the hypothesis G has an edge e = {a, b} with a ∈ A and b ∈ B such that h(b) − h(a) = r. We take j = q + 1, which yields hj (b) − hj (a) = h(b) + (j − 1)n − h(a) = qn + r = i. Thus H is a simple graph. Now we claim that hj is one-to-one on V (G) for 1 ≤ j ≤ x. It is clear that hj is one-to-one on each set A and B since h is one-to-one. Furthermore, max{hj (a) : a ∈ A, 1 ≤ j ≤ x} = max(h(A)) < min(h(B)) = min(h1 (B)) = min{hj (b) : b ∈ B, 1 ≤ j ≤ x}. Thus for fixed j, 1 ≤ j ≤ x, the edges {hj (u), hj (v)} as {u, v} runs through E(G) form an isomorphic copy of G. Thus G divides H. Furthermore, taking each vertex of H as its label gives an α-labeling of H, producing a cyclic H-decomposition of K2nx+1 , and thus a cyclic G-decomposition of K2nx+1 . The strategy for generating a G-decomposition of K2nx+1 from an α-labeling is illustrated in Figure 3.1. 3.1.1. Ordered labeling for bipartite graphs An immediate shortcoming of the α-labeling result in Theorem 3.1 is the requirement (as a consequence of the definition for α-labelings) that the labeling be graceful. Hence, any Eulerian graph that admits an α-lableling must satisfy the Parity Condition (Theorem 1.2). This rules out getting an α-labeling for any even cycle C2m where m (at least 3) is odd. Furthermore, Rosa noted in [51] that not all trees, which are conjectured to all be graceful, admit an α-labeling..
(38) 23. 3.1. Generalization of α-labeling. 0. 1. 3. 0. 1. 3. 0. 7. 2. 6. 7. 2. 6. 19 14 18. +. (a) ρ -labeling of C6. 1. 3. 0. 1. 3. 31 26 30. (b) Three starters for a cyclic (K37 , C6 )-design. Figure 3.2: An ordered ρ-labeling of a 6-cycle and the 3 starters obtained by way of Theorem 3.2 for a cyclic C6 -decomposition of K37 .. In an effort to pick up where the α-labeling left off, we now present the following result on ordered ρ-labelings as shown by El-Zanati, Vanden Eynden, and Punnim [28].. Theorem 3.2. Let G be a bipartite graph with n edges. If G admits a ρ+-labeling, then there exists a cyclic G-decomposition of K2nx+1 for all positive integers x.. Given that this result is subsumed by Theorem 3.5, we omit a full proof of Theorem 3.2 here, but instead describe the overall strategy (which is extended for the proof seen later for Theorem 3.5): Let bipartite graph G with n edges and vertex bipartition {A, B} admit ρ+-labeling h and let B1 , B2 , . . . , Bx be x unique copies of B. Construct x copies of G, i.e. G1 , G2 , . . . , Gx such that Gi has vertex bipartition {A, Bi }. Then the labeling function h(v) v ∈ A, ˜ h(v) = h(v) + (i − 1)2n v ∈ B i. S is a ρ-labeling of the graph xi=1 Gi (in fact, it is itself a ρ+-labeling of that larger graph). This strategy for generating a G-decomposition of K2nx+1 from an ordered ρ-labeling is illustrated in Figure 3.2. 3.1.2. Labelings for tripartite graphs In [16], this author and others generalized the concept of a γ-labeling for tripartite graphs removing the requirement that the graph be almost-bipartite. Specifically, two new labelings (one of them subsuming γ-labelings) were introduced, and it was shown that if a tripartite graph G with n edges admits one of these labelings, then there exists a cyclic G-decomposition of K2nx+1 for any positive integer x..
(39) 24. 3.1. Generalization of α-labeling 0. 1. 1. 8 5 7. A. 3 4 6. 8. 2. 2. C. 5. 3. B. Figure 3.3: A σ-tripartite labeling of a graph G with 8 edges.. σ-tripartite Labelings Let G be a tripartite graph with n edges having the vertex tripartition {A, B, C}. A σ-tripartite labeling of G is a one-to-one function h : V (G) → [0, 2n] that satisfies the following conditions: (s1) h is a σ-labeling of G; (s2) if {a, v} ∈ E(G) with a ∈ A, then h(a) < h(v); (s3) if e = {b, c} ∈ E(G) with b ∈ B and c ∈ C, then there exists an edge e0 = {b0 , c0 } ∈ E(G) with b0 ∈ B and c0 ∈ C such that |h(c0 ) − h(b0 )| + |h(c) − h(b)| = n; (s4) if a ∈ A and v ∈ B ∪ C, then h(a) − h(v) 6= n; (s5) if b ∈ B and c ∈ C, then |h(b) − h(c)| 6∈ {n, 2n}. Note that e and e0 in (s3) need not be distinct. Figure 3.3 shows a σ-tripartite labeling of a graph G with 8 edges. We also note that there are tripartite graphs that do not admit σ-tripartite labelings. For example, if in each tripartition of V (G) the number of edges between each pair of vertex sets in the tripartition is odd (as in C3 ), then G has no σ-tripartite labeling. Theorem 3.3. If a tripartite graph G with n edges admits a σ-tripartite labeling, then there exists a cyclic G-decomposition of K2nx+1 for each positive integer x. Proof. Let h be a σ-tripartite labeling of G, with sets A, B, and C as in the definition. Since G has no isolated vertices, h(A) < 2n. We can assume x > 1 by Theorem 1.1..
(40) 25. 3.1. Generalization of α-labeling. For 1 ≤ j ≤ x define hj : V (G) → [0, 2nx] by h(v) v ∈ A, hj (v) = h(v) + (j − 1)n v ∈ B, h(v) + (x − j)n v ∈ C. We define a multigraph H with vertex set contained in [0, 2nx] and with the nx edges {hj (u), hj (v)}, one for each edge {u, v} of G and 1 ≤ j ≤ x. We will show that the set of labels |hj (u) − hj (v)| of edges in H is exactly {1, 2, . . . , nx}, so that actually H is a simple graph, without loops or multiple edges. Suppose G has an edge {s, t} between B and C. For 1 ≤ j ≤ x, let g(j) = (j − 1)n − (x − j)n = (2j − x − 1)n, and note that g(x + 1 − j) = −g(j). Define k = k(s, t, j) to be j or x + 1 − j according as s ∈ B or s ∈ C. Then in any case there exists k, 1 ≤ k ≤ x, such that hk (s) − hk (t) = h(s) − h(t) + g(j).. (3.1). Now let 1 ≤ i ≤ nx. We will show that H has an edge with label i. The proof will be by cases depending on q and r, where q and r are integers such that i = qn + r,. 1 ≤ r ≤ n,. 0 ≤ q < x.. In the proof we will use vertices hj (v), where v ∈ V (G). In all cases it can be checked that j is an integer and 1 ≤ j ≤ x. By assumption G has an edge e = {u, v} such that h(u) − h(v) = r.. Case 1: v ∈ A. If u ∈ B then take j = q + 1. Then hj (u) − hj (v) = h(u) + (j − 1)n − h(v) = qn + r = i. If u ∈ C then take j = x − q. Then hj (u) − hj (v) = h(u) + (x − j)n − h(v) = qn + r = i..
(41) 26. 3.1. Generalization of α-labeling 0. 1 1. 24 21 23. 0. 13. 20. 15. 7. 13. 17 5. 12. 19 4 10. 10 8. 2. 3. 1. 8. 11 6. 2 16. 18. 21. 0 9. 16. 3 22. 24. 1. 18. 14. 11. 5. 19. Figure 3.4: The three copies of G (from Figure 3.3) obtained by way of Theorem 3.3 for a cyclic G-decomposition of K49 .. Case 2: {u, v} ⊆ B ∪ C and q ≡ x + 1 (mod 2). Take j = (q + x + 1)/2. Then for some k hk (u) − hk (v) = h(u) − h(v) + g(j) = r + (2j − x − 1)n = r + qn = i. Case 3: {u, v} ⊆ B ∪ C and q ≡ x (mod 2). By condition (s3) then G has an edge {u0 , v 0 } ⊆ B ∪ C such that h(u0 ) − h(v 0 ) = n − r. Take j = (x − q)/2. Then |hk (u0 ) − hk (v 0 )| = |h(u0 ) − h(v 0 ) + g(j)| = |n − r + (2j − x − 1)n| = | − r − nq| = i. Thus H is a simple graph. Now we claim that hj is one-to-one on V (G) for 1 ≤ j ≤ x. It is clear that hj is one-to-one on each set A, B, and C. First we show that hj is one-to-one on A ∪ B. Suppose hj (a) = hj (b), a ∈ A, b ∈ B. Then h(a) = h(b) + (j − 1)n. Clearly j > 1 since h is one-to-one. If j > 2, then h(a) < 2n ≤ h(b) + 2n ≤ hj (b), so we must have j = 2, and this case is excluded by condition (s4). A similar proof shows that hj is one-to-one on A ∪ C. Finally we show hj is one-to-one on B ∪ C. Suppose hj (b) = hj (c), b ∈ B, c ∈ C. Then h(c) − h(b) = (2j − x − 1)n. Clearly |2j − x − 1| must be 1 or 2. This contradicts condition (s5). Thus for fixed j, 1 ≤ j ≤ x, the edges {hj (u), hj (v)} as {u, v} runs through E(G) form an isomorphic copy of G. Thus G divides H. Furthermore, taking each vertex of H as its label gives a σ-labeling of H, producing a cyclic H-decomposition of K2nx+1 , and thus a cyclic G-decomposition of K2nx+1 . Figure 3.4 shows the three copies of the graph G (from Figure 3.3) that can be used to produce a cyclic G-decomposition of K49 ..
(42) 3.1. Generalization of α-labeling. 27. It is easy to see that if a graph G with n edges admits a σ-labeling, then there exists a cyclic G-decomposition of K2n+2 − I, where I is a 1-factor in K2n+2 . (The edges of length n + 1 constitute the 1-factor in K2n+2 .) Since the graph H in the proof of Theorem 3.3 admits a σ-labeling, we have the following corollary. Corollary 3.4. Let G be a graph with n edges that admits a σ-tripartite labeling and let x be a positive integer. Then there exists a cyclic G-decomposition of K2nx+2 − I, where I is a 1-factor of K2nx+2 . ρ-tripartite Labelings Let G be a tripartite graph with n edges having the vertex tripartition {A, B, C}. A ρ-tripartite labeling of G is a one-to-one function h : V (G) → [0, 2n] that satisfies the following conditions: (r1) h is a ρ-labeling of G; (r2) if {a, v} ∈ E(G) with a ∈ A, then h(a) < h(v); (r3) if e = {b, c} ∈ E(G) with b ∈ B and c ∈ C, then there exists an edge e0 = {b0 , c0 } ∈ E(G) with b0 ∈ B and c0 ∈ C such that |h(c0 ) − h(b0 )| + |h(c) − h(b)| = 2n; (r4) if b ∈ B and c ∈ C, then |h(b) − h(c)| = 6 2n. Note that e and e0 in (r3) need not be distinct. Figure 3.5 shows a ρ-tripartite labeling of the Petersen graph P . We note that although a σ-labeling is necessarily a ρ-labeling, a σ-tripartite labeling is not a ρ-tripartite labeling. We also note that a γ-labeling is necessarily a ρ-tripartite labeling. We also note that there are tripartite graphs that do not admit ρ-tripartite labelings. For example, if in each tripartition of V (G) the number of edges between each pair of vertex sets in the tripartition is odd (as in C3 ), then G has no ρ-tripartite labeling. Theorem 3.5. If a tripartite graph G with n edges admits a ρ-tripartite labeling, then there exists a cyclic G-decomposition of K2nx+1 for each positive integer x. Proof. Let h be a ρ-tripartite labeling of G, with sets A, B, and C as in the definition. Since G has no isolated vertices, h(A) < 2n. We can assume x > 1 by Theorem 1.1. Let B1 , B2 , . . . , Bx and C1 , C2 , . . . , Cx be x vertex-disjoint copies of B and C, respectively. The vertices in Bi and Ci corresponding to b ∈ B and c ∈ C will be.
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