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Problems Column 217

Discussiones Mathematicae Graph Theory 25 (2005 ) 217–218

ESTIMATION OF CUT-VERTICES IN EDGE-COLOURED COMPLETE GRAPHS

Adam Idzik Akademia ´Swi¸etokrzyska

15 ´Swi¸etokrzyska street, 25–406 Kielce, Poland and

Institute of Computer Science Polish Academy of Sciences 21 Ordona street, 01–237 Warsaw, Poland

e-mail: adidzik@ipipan.waw.pl

All graphs considered here are finite simple graphs, i.e., graphs without loops, multiple edges or directed edges. For a graph G = (V, E), where V is a vertex set and E is an edge set, we write sometimes V (G) for V and E(G) for E to avoid ambiguity. We shall write G \ v instead of GV \{v} = (V \ {v}, E ∩ 2V \{v}), the subgraph induced by V \ {v}. A vertex v ∈ V (G) is called a cut-vertex of G if G is connected and G \ v is not. By a k-edge- colouring of a graph we mean any finite partition of the set of its edges into k subsets. A graph (V, E) with a given k-edge-colouring (E1, · · · , Ek) (Ei ∩ Ej = ∅ for i 6= j; i, j ∈ {1, · · · , k} and Si∈{1,···,k}Ei = E) is denoted by (V, E1, · · · , Ek). The graphs (V, Ei) are called monochromatic subgraphs of (V, E1, · · · , Ek), i ∈ {1, · · · , k}. As usual, by Km we denote the complete graph with m vertices.

Let c(Gi) denote the number of cut-vertices of Gi in a monochromatic subgraph Gi = (V, Ei) of a k-edge-coloured complete graph Km = (V, E1,

· · · , Ek) (i ∈ {1, · · · , k}).

Given a k-edge-coloured graph G = (V, E1, · · · , Ek), we define Fi = E \ Ei, Gi = (V, Ei), ¯Gi = (V, Fi), where E =Si∈{1,···,k}Ei and i ∈ {1, · · · , k}.

Here Gi is a monochromatic subgraph of G and ¯Gi its complement in G.

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218 A. Idzik

Theorem (Idzik, Tuza, Zhu). Let (E1, · · · , Ek) be a k-edge-colouring of Km (k ≥ 2, m ≥ 4), such that all the graphs ¯G1, · · · , ¯Gk are connected.

(i) If one of the subgraphs G1, · · · , Gk is 2-connected, say Gi, then c( ¯Gi) ≤ m − 2 and c( ¯Gj) = 0 for j 6= i (i, j ∈ {1, . . . , k}).

(ii) If none of the graphs G1, · · · , Gk is 2-connected, and one of them is connected, say Gi, then c( ¯Gi) ≤ 2 (i ∈ {1, · · · , k}).

(iii) If none of the graphs G1, · · · , Gk is 2-connected, and one of them is disconnected, say Gi, then c( ¯Gi) ≤ 1 (i ∈ {1, · · · , k}).

Problem. Let (E1, · · · , Ek) be a k-edge-colouring of Km (k ≥ 2, m ≥ 4).

What is the cardinality of the set of the sum of cut-vertices of ¯Gi in the case none of Gi is 2-connected and (a) two of Gi are connected or (b) two of Gi are disconnected and c( ¯Gi) = 1 (i ∈ {1, · · · , k}) ?

Observe that in both cases (a) and (b) all the graphs ¯G1, · · · , ¯Gk are con- nected.

This problem is related to some theorems presented in [1] and [2].

References

[1] J. Bos´ak, A. Rosa and ˘S. Zn´am, On decompositions of complete graphs into factors with given diameters, in: P. Erd˝os and G. Katona, eds., Theory of Graphs, Proceedings of the Colloquium Held at Tihany, Hungary (Academic Press, New York, 1968) 37–56.

[2] A. Idzik and Z. Tuza, Heredity properties of connectedness in edge-coloured complete graphs, Discrete Math. 235 (2001) 301–306.

Received 21 November 2003

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