On the locally Hamiltonian graphs and Kuratowski’s theorem

ANNALES SOCIETATIS MATIIEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

Z. S

(Kraków)

On the locally Hamiltonian graphs and Kuratowski’s theorem

1. Introduction. This paper contains the proofs of some propositions announced in [3] p). We will use the definitions of [3]. Notation and further definitions are given in § 3.

The author has shown in [4] that the collection of the connected, locally Hamiltonian graphs with n vertices and 3n — 6 edges coincides with the collection of the 8 2 triangulation graphs with n vertices, and that a connected and locally Hamiltonian graph with n vertices contains at least 3n — 6 edges and is non-planar if and only if it contains more than 3n — 6 edges. The famous theorem of Kuratowski [2] says that a graph is non-planar if and only if it contains topologically either K 5 or К (3,3). From the results of the present paper it follows, however, that a non-planar locally Hamiltonian graph contains topologically K s and if it is connected and the number of its vertices is different from 5 then it contains topologically also К (3,3). Owing to these results we can for­

mulate certain characterizations of 8 2 triangulation graphs (see [3], th. 8).

From Theorem 5 below it follows that the property of the 1-skeleton of a triangulation of a given surface (i.e. compact two-manifold or two- manifold with boundary) to be planar or not, is a topological property of the surface.

2. Results. Let us recall some definitions. Given a non-isolated vertex ж of a graph G, G(x) denotes a subgraph of G spanned by the set of all vertices adjacent to x in G. A graph G is Hamiltonian (almost Hamiltonian) if it has a Hamiltonian circuit (path), i.e., a circuit (path) which passes through each vertex of G. G is locally (almost) Hamiltonian if the graph G(x) is non-empty and (almost) Hamiltonian for every vertex x of G.

T

1. A connected and locally Hamiltonian graph G is 3-con­

nected.

(x) By mistake the assumption that n > 3 in the statement of th. 9 in [3] was

omitted.

L

1. Let T be a triangulation of a compact two-manifold M with Euler characteristic % < 2 and let G be the 1-skeleton of T. Given a vertex x of G, let 8 (x , T) be a circuit of G which covers the boundary of the open star of x in T. Then:

(a) There is in G a regular circuit К which does not divide M.

(b) For any vertex x of K , there exists a circuit L = L (x , К , T) of G containing x , no other vertex of К and exactly two vertices of 8 (x, T) sepa­

rated in 8 (x , T) by vertices adjacent to x in K.

(c) К contains a vertex x 0 of degree d(x0, G) > 5.

T

2. The 1-skeleton G of a triangulation T of a compact two- manifold topologically different from the two-sphere 8 2 contains topolo­

gically both K 5 and К (3,3).

L

2. I f a locally Hamiltonian graph G does not contain topolo­

gically K 5, then, for any three vertices u, w, у of G, the following implication is valid:

I f и and у are adjacent in a Hamiltonian circuit of G(w), then w and у are adjacent in any Hamiltonian circuit of G(u).

C

1. For any vertex и of a locally Hamiltonian graph G which does not contain topologically K 5, there is a unique Hamiltonian circuit of G(u).

T

3. A connected and locally Hamiltonian graph G, which contains no subgraph homeomorphic to K 5, is an S 2 triangulation graph.

T

4. Each connected, locally Hamiltonian and non-planar graph G with n > 5 vertices contains topologically К (3,3).

E e m a r k l. If 6r is a connected and locally Hamiltonian graph with n vertices, then n > 4, G contains at least 3n — 3 edges, and G is non-planar if and only if it contains more than 3n — 6 edges (cf. [4]). Consequently, by the above results, we have:

C

2. A connected, locally Hamiltonian graph G with n vertices contains topologically K 5 (.К (3,3)) if and only if G has at least 3n — 5 edges (and n > 5).

D

. A circuit J of a connected graph G is said to be peri­

pheral in G if either J — G or the removal of J from G yields an open sub­

complex of G which is connected (i.e., is not the union of two non-empty disjoint open subcomplexes of G).

R em ark 2. Each peripheral circuit of a connected and planar graph G covers the boundary of a face of any realization of G on the two-sphere 8 2 (cf. [1], p. 825 or [5], p. 748).

L

3. I f the 1-skeleton G of a triangulation T of a surface M (i.e.,

compact two-manifold or two-manifold with boundary) is 3-connected, then

the poundary complex of any two-simplex of T is a peripheral circuit of G.

T

5. Let M be a compact two-manifold or two-manifold with boundary. Then a necessary and sufficient condition that M be homeomorphic with a subset of $ 2 is that there is a triangulation of M whose l-sheleton is planar.

3. Further definitions and notation. Only finite graphs without loops and multiple edges will be considered. Thus, by a graph we will mean a finite simplicial zero- or one-complex. Throughout, “complex”

will refer to a finite simplicial complex (abstract or concrete). We will deal with complexes and simplexes which may be topological (curvi­

linear).

Two complexes K x, K 2 are said to be isomorphic if there is an 1-1 mapping cp: A x -> A 2 from simplexes of K x onto those of A 2, called isomorphic transformation, which preserves dimensions and incidences.

A realization or an imbedding A 0 of a complex A in a topological space X is a complex isomorphic to К with all simplexes in X . Point-set union of all simplexes of K 0 is a polytope and is denote by |A0| ; K 0 is a trian­

gulation of \K9\. If A 0 is a graph, then each component of X \ |A 0| is called a face of A 0 in X , and it is said that A 0 does not divide (separate) X if there is but one face of K 0 in X . It is known that any two polytopes

\KX\ and \K2\ with isomorphic finite triangulations K x and K 2, respecti­

vely, are homeomorphic. Two complexes K x and K 2 are called homeo­

morphic (or one of them is said to be a homeomorph of the other) if there is a polytope with two triangulations: one isomorphic to K x, the other to K 2. Thus, two polytopes with homeomorphic triangulations are homeo- morpliic.

An 1 -sheleton or a graph of a complex A is a subcomplex of К con­

sisting of all vertices and edges of K. Given a vertex x of K , a star (closed star) of ж in К is the collection of all simplexes of К of which a; is a face (and all their faces). An open star of ж in a concrete complex К is the point-set union of all elements of the star of ж in K .

Given a complex K , we write x e K only if ж is a vertex of K , and we write P ę К if P is a subcomplex of K.

A degree of a vertex ж of a graph G is the number of edges of G incident to x, and is denoted by d(x,G ). A vertex ж of G is called a node of G if d(x, G) ф 2. A path is a graph isomorphic to a triangulation of a closed line interval, and a circuit is a triangulation of a simple closed curve.

A circuit A of a graph G is regular in G if any two vertices of A adjacent in G are adjacent in A too.

A length l(P) of a path (circuit) P is the number of its edges. A path P of length l(P) = 1 and with nodes y ,z (уф я) is denoted by [y,z], and is called an edge or more exactly an edge-graph. The sum Gx-\-G 2 of two graphs Gx and G 2 is a graph whose set of vertices is the union of those of

Roczniki PTM — Prace M atematyczne XI.2 17

Gx and G2, and two vertices are adjacent in Gx + 6r2 if and only if they are adjacent either in Gx or in G2.

The symbol [xl r x2, ..., xn\ will denote the snm [Ai , x 2 j [^2 j • • • ~b \jfin— 1 j ^nj

if n > 2, Xi Ф Xj for i j j = 1, 2, ..., n, i Ф j, and, for n > 4, {i, j}

Ф {1, n). If x x Ф xn, then x x, xn are the only nodes of P = [xx, x 2, ..., xn] , and P is a path with end points x x and x n; and the remaining vertices (if any) of P are called internal vertices of P.

We say that a path connects or joins its end points. Given a path P which contains the vertices у and z, the portion of P which joins у and z is denoted by P [ y , я]. If у — z, then P [y , я] means simply the vertex y\

if у and z are end points of P, then P[y,z~] — P. Let V denote the set of vertices of a graph G. Given a set U = {xx, x 2, ..., xk}, the symbol G \( x x, x 2, xk) denotes a subgraph of G spanned by the set V \U .

A complex К is connected if it is not the union of two non-empty disjoint subcomplexes either both closed or both open in K. It is known that К is connected if and only if any two of its vertices can be joined by a path of the 1-skeleton of K . It is said that the removal of some ver­

tices of К does not disconnect К if any two of the remaining vertices can be joined in the 1-skeleton of A by a path which contains no removed vertex. Given an integer h > 1, a graph G is said to be h-connected if it contains at least h -\-l vertices and the removal of any h — 1 vertices does not disconnect G.

4. Proofs. P r o o f of T heorem 1. Since G is locally Hamiltonian, it contains at least 4 vertices. So we need only point out that, for any vertices x x and x 2 of G, the graph G \{ x x, x 2) is connected. To prove this, let у and 0 be any two different vertices of G \( x x, x 2). We will show that there is a path of G \( x x, x 2) with end points у and 0 . Since G is connected, there is a path P of G connecting у and 0 . If X iiP , i = 1 ,2 , then P £ G \( x x, x 2) . Let us assume, without loss of generality, that x xeP. Hence there exist in P two vertices t and и adjacent to x x in G and such that each of the portions P [ y ,t] and P [u, 0 ] of P contains no other vertex of G adjacent to x x. The subgraph G(xx) of G is Hamiltonian. So there exists in G(xx) a path P x that connects t and и and whose each internal vertex is different from x2. The graph Q = P [ y , t] - fP x-j-P[u, z] is a path of G which connects у and z, and does not contain x x. It is easily seen that if x 2 4 P, then

24 Q and Q £ G \{ x x, x 2). If x 2 eQ, then similarly as above one can construct a new path connecting у and z, and included in G \( x x, x 2) . Thus the theorem is proved.

P ro o f of Lem m a 1. (a) It is known that if % < 2, G contains

a circuit K 0 not dividing M. If K Q is not regular in G, there exist in K 0

two vertices x x and x 2 which are adjacent in G and divide K 0 into two paths P

= Pj[aj1,®2] and P'0' — P^ {xx, x 2] both of length > 2 . The graphs K '0 = P' 0 + [ x x, x 2] and K'0' = P'0'+ [xx, х 2] are circuits of G.

Each of their lengths l(K'0) and l{K'o) is less than l(K 0). Since K 0 does not divide M, at least one of circuits K'Q and K'0' does not either. Denote this circuit by K x.

If the circuit К x is not regular in G, similarly as above One can obtain a new circuit K 2 which does not divide M and has the length 1{K2) < l(K x) . Since every circuit of length 3 is regular, after a finite number, r, of steps one can get a regular circuit К , К = K r, not dividing M. Thus, Lemma 1 (a) is proved.

(b) Let y x, y 2 be all the vertices of К adjacent to x. There exist in T two triangles (i.e., closed two-simplexes) A and A 0 with vertices x , y x, и and x, y x, u 0 (u, u 0 ф х , у х and и ф и 0), respectively. Since К does not divide M, there is a chain of triangles A0, A x, ..., Ap — A such that any two successive triangles have a common edge not included in K . Further, by the properties of K , Ai (i — 0 , 1 , . . . , p) contains at most two vertices included in К and, if this is the case, the edge incident to both of them is included in К too. Hence и, u 0 4 K , u 0 eAx, and there is an integer 7 such that 1 < 7 < p, u0eAi and u (X 4 Ai for % = 7 + 1 ,

Thus, the edge of Аг not incident to u 0 is contained in Аг+Х and therefore it is not included in K . Hence there is a vertex u x such that u xeAi{i

— 1,1+ 1), u x Ф u 0 and uxĄK. Obviously, u 0 and ux are adjacent in G.

Now, if u x Ф u, utilizing a similar procedure, we can find a next vertex u 2 adjacent to u x and not included in and such a step-by-step con­

struction provides a path P — [u0, ux, ..., un — u \ which is included in G and contains no vertex of K .

Obviously, and и are separated in 8 (x , T ) by the vertices y x, y 2 adjacent to x in K . So P contains two different vertices of 8 {x, T), t 0 and t say, such that y x and y 2 separate t 0 from t in 8 (x, T) and no internal vertex of the portion P [t, tf0] of P is adjacent to x. The graph P [t, £0] + + [t, x, t0] is a circuit of G and, clearly, we may take it to be L (x , К , T ), as is required in Lemma 1(b).

(c) From (b) it follows that each vertex of К is of degree not less than 4 in G. Suppose that each vertex of К is of degree 4 in G. So, for a vertex x of K , there exist exactly two vertices t and t 0 not included in К and adjacent to x in G, and each of two neighbours y x and y 2 of x in К is adjacent both to t and to t0. Thus t and t 0 are adjacent to each vertex of K , and no other vertex not contained in К is adjacent to a vertex of K . Since, by Theorem 1, G is 3-connected, it contains no vertices different from t, tQ and those of K. Therefore G contains exactly a 0 = 7(71)+ 2 verti­

ces and aj edges where

(1)

3l(K) + l = 3a0— 5,

if t and Z0 are adjacent in G or

(2) ax == За0— 6,

if they are not adjacent in G. On the other hand, by Euler’s formula (see also [3], Corollary 1), G contains 3a0 — 3# edges where, by assumption, X < 1. This contradicts both (1) and (2).

P ro o f of T heorem 2. Using the notation of Lemma 1, we see that for any vertex x of K , the graph К -f L (x , К , T) + 8 (x , T) is a homeo- morph of K $ in G.

Recall that x QeK and <Z(a?0, G) > 5. Let г be a vertex adjacent to x 9 in G and such that z j K , z 4 L (x 0, K ,T ) . It is easy to see that the graph K-\- L Ą - 8 Ą- [a?0, «] is a subgraph of G which contains a homeomorph of К ( 3, 3) in G. This subgraph is shown in Fig. 1 (where x ~ ж0).

P r o o f of L em m a 2. Suppose, if possible, that there exist in G three vertices u ,w ,y and Hamiltonian circuits H [u] and H[w] of G(u) and G(w), respectively, such that и and у are adjacent in H[w], and w and у are not adjacent in H [u]. Hence у and two neighbours of w in H [u] are adjacent in G both to и and to w. Therefore there exist l vertices (Z>3) adjacent in G both to и and to w. Let them be Zx, t2, . . . , ^ = у in the cyclic order in which they appear in H[w]. Since w and у are not adjacent in H [u], they divide H [u ] into two paths and P 2 (Px =

= P 1 [y,w'}, P 2 = p 2ly,w ], Р г+ Р 2 = H[u]) with lengths Z(P1),Z(P2) both

> 2, where the notation is chosen so that t 1 €P1. Hence there is an integer j, 1 < j < Z, such that [w, Z,-] c: P 2. Therefore there is the smalest integer Tc, 1 < к < j, such that tk ^P 1 and tk+ 1 eP2. Let Q be a path of H[w] with end points tk and tk+l and such that у 4 Q. Now, since и Ф tk, t k+1 and [u , y ]

H[w], it follows that u^Q. Put В г = P x[y, tk] and P 2

= P 2 [y,Zfc+1]. It is easy to see that each of the paths Q ,R l , B 2 contains

neither и nor w, and that any two of these paths can have only their

end points in common. Thus G contains a subgraph Q + P x + P 2 +

+ [u, tk, w, u] + \_u i h+i, w, У, homeomorphic to K 5, contrary to the hypothesis. This subgraph is shown in Fig. 2.

P ro o f of C orollary 1. By Lemma 2, for any vertex w of G(u), both of the neighbours of и in a Hamiltonian circuit of G(w) must be adjacent to w in any Hamiltonian circuit of G(u). Therefore there is only one Hamiltonian circuit of G{u).

P roof of T h eorem 3. Let T be a two-dimensional simplicial complex whose 1-skeleton is G and such that there is in Г a two-simplex with vertices u, w, у if and only if any two of these vertices, и and у say, are adjacent in a Hamiltonian circuit of G(w). By Lemma 2 and Corollary 1, the open star of any vertex of T is a two-cell and, furthermore, T is con­

nected because so is G. Thus, T is a triangulation of a closed two-manifold.

Let % be its Euler characteristic. Supposition that % < 2 contradicts Theorem 2. Therefore x — 2. Hence G is an $ 2 triangulation graph, and the proof is complete.

P ro o f of T heorem 4. Owing to the well-known theorem of Kura- towski [2], we need only prove that if G contains topologically K 5, then G has a subgraph homeomorphic to К (3,3).

Let F be a subgraph of G homeomorphic to K 5. Given a node x of F, we can find a homeomorph F x of K 5 in G whose one node is x and the remaining ones are adjacent to ж in Let xx, x 2, x 3, x 4 be all the nodes of F different from x, and let P i '(i = 1 ,2 ,3 ,4 ) be a path of F which

О О

connects x and x 4 and contains no node of F distinct from x and x4. Simi-

_ О О

larly, let Pij (1 < i, j < 4, i Ф j) be a path of F which connects Xi and Xj,

and does not contain any node of F different from x

4 and Xj. Each of

Pi (i = 1 , 2 ,3 ,4 ) contains a vertex xt (possibly x 4 = х4) which is adjacent

to x in G and is the only vertex of Рг[х4, xt] adjacent to x in G. Let Рц

— P i[х{, Xi]+ P%+ Pj [Xj, Xj], 1 < i, j < 4, i Ф j. The paths Рц and Р кг are disjoint if {i, j , Tt, 1} = { 1 , 2 ,3 ,4 } . Suppose, without loss of generality, that x x and x 3 separate x 2 from xx in a certain Hamiltonian circuit 8 of 0{x). Therefore P 24 contains two vertices of 8 , y x and у 2 say, such that they separate x x from x 3 in 8 and are the only vertices of the circuit К — [y2, x , у x~\ P 24 \_y \ j Уъ\ which are adjacent to x in O. Analo­

gously, P 13 contains two vertices of 8 , t 0 and t say, which are separated by yx and y 2 in 8 and such that they are the only vertices of the circuit L — [t, x , t 0 ]J\-Pls[t0, f\ which are adjacent to x in G. It is easily seen that the graph F x = S-Ą-K-Ą-L is a homeomorph of K 5 in G, as is required.

The vertex x may be called a centre of F x.

Next, we will prove that there exists a node of F x with degree at least 5 in G. Let us assume the contrary. Since G is connected and contains at least 6 vertices, F x contains a vertex z which is adjacent to one of the nodes of F x and is of degree 2 in F x. By assumption, z can not be adjacent to x in ( t ; hence z j S . Suppose, without loss of generality, that z is adjacent to t. Therefore zeP 13 [t0, t]. It is easy to see, however, that t is isolated in G{z); hence G(z) is not Hamiltonian, contrary to the assumption that G is locally Hamiltonian.

Thus, there is a node v of F x which is of degree at least 5 in G. If v = x,G contains such a subgraph as is shown in Fig. 1, whence G contains topologically K ( 3,3). If v Ф x, we apply the above construction to obtain a homeomorph of K 5 in G with the centre v. So G contains topologically К (3,3) in this case too. Thus Theorem 4 is proved.

P r o o f of Lem m a 3. Let A be a two-simplex of T, and let x u x 2, x 3 be all the vertices of A. Since G is 3-connected, then G is connected, contains at least four vertices and is not any circuit; and each graph of the form G \(X i, Xj) (i, j = 1 ,2 ,3 ) is connected. Thus, we need only show that G \( x x, x 2, x 3) is connected. Two cases may arise:

(a) No edge of A is situated on the boundary of M.

(P) An edge of A is on the boundary of M.

In case (a), T contains a subcomplex shown in Fig. 3, where possibly ti — tj for i Ф j ( i,j = 1 ,2 ,3 ) . G is clearly locally almost Hamiltonian.

Therefore, in order to prove that G \( x x, x 2, x 3) is connected, it suffices to show that ti , t 2, t 3 are in the same component of G \ ( x x, x 2, x 3). It is easy to see that if G(x 1 ) \ ( x 21 x 3) is not connected, it consists of two com­

ponents one of which contains t 2 and the other contains t3. Since the graph G \( x 2, x 3) is connected, it contains a path joining tx and х г. This path contains a vertex of G(x 1 ) \ ( x 2, x 3); and therefore tx and t 2 or tx and t 3 are in the same component of G \( x x, x 2, x 3). Suppose, without loss of generality, that tx and t 2 are in the same component of G \( x x, x 2, x 3) . G \( x t , x 2) is clearly connected and contains a path connecting t 3 and x 3.

By the similar argument as above, t 3 and tx or t 3 and t 2 are in the

same component of G \( x x, x 2, x 3), whence, by the preceding supposition, tx, t 2 , t 3 are in the same component of G \( x x, x 2, x 3) . Thus, in case (a) the proof is complete.

Assume that the edge connecting х г and x 2 is situated on the bound­

ary of M. Neither of the remaining edges of A can be on the boundary

of M \ for otherwise G would contain a vertex of degree 2 and therefore would not be 3-connected. Now, T contains a subcomplex shown in Fig. 4 where possibly tx = t2. Since G \( x x , x 3) (as. well as G \{ x 2, x3)) is connected, tx and t 2 are in the same component of the graph G \( x x, x 2, x 3). Thus, this graph is connected and the lemma is proved.

P ro o f of T heorem 5. The necessity of the condition is obvious.

To prove the sufficiency, let T* be a triangulation of M with a planar 1-skeleton G*. If M is a compact two-manifold with Euler characteristic and if G* is the graph with n vertices then, by Euler’s formula, G*

contains 3n — 3% edges. On the other hand, G* is a connected, locally Hamiltonian and planar graph which, by Eemark 1, contains 3n — 6 edges. Hence % — 2, and M is homeomorphic to $ 2.

Now, let M be a two-manifold with boundary, and assume that it is non-planar (i.e., topologically different from any proper subset of $ 2). G* is then a connected and locally almost Hamiltonian graph. It is easily seen that the removal of any vertex of T* does not disconnect G*, and that the removal of any two vertices, x and у say, can disconnect G* only if x and у are adjacent in G*. If it is the case, the closed edge connecting x and у divide M into two pieces. Denote their closures by M x and M 2, respectively. Each of M x and M 2 may be regarded as a new two-manifold with boundary and with the triangulation which is a subc'omplex of T*.

Since M is non-planar, one of M x, M 2 is so too. Suppose that M x is non- planar. Let Т г be the triangulation of M x such that T x <=. T*, and let Gx be the graph of T x) hence Gx c~G*. If Gx is not 3-connected, we can continue the process, i.e., we disconnect M x by the removal of an appro­

priate closed edge of Gx, etc. Since G* is a finite graph, we arrive in the end

Fig. 3 Fig. 4

at a non-planar two-manifold with boundary, N say, with a triangulation T whose 1-skeleton, G say, is a 3-connected subgraph of G*. Hence G is planar, and T cannot be imbedded in 8 2.

Let G 0 be a realization of G in the two-sphere 8 2, and let cp: G -+G 0 be an isomorphic transformation of G onto GQ. We will extend this trans­

formation so as to obtain an isomorphic transformation, still called <p, of T onto a certain complex situated in 8 2. Thus, we will obtain a con­

tradiction. Let us first note that G 0 is not any circuit. Hence, by Lemma 3 and Eemark 2, given an open two-simplex A of T, there is exactly one face of the graph G 0 in 8 2, <p(A) say, whose boundary complex is such a circuit of G

which consists of the images under q> of all elements of the boundary circuit of A. Let T 0 be the collection of all <p(A) and all elements of G0. Of course, T 0 is a two-complex, and <p: T T 0 is an isomorphism. Thus, the theorem is proved.

A d d e d i n p r o o f . It is worthy of note in connection with Corollary

(see also [

] and [3], corollary

and th. 1) the following conjecture of Dirac: Each graph with n vertices and at least Ъп — 5 edges contains topologically K

.

R eferences

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Math. 15 (1930), pp. 271-283.

[3] Z. S k u p ie ń , Locally Hamiltonian graphs and Kuratowski theorem, Bull.

Acad. Polon. Sci., Ser. sci. math., astr. et phys. 13 (1965), pp. 615-619.

[4] — Locally Hamiltonian and planar graphs, Fund. Math. 58 (1966), pp. 193-

-200^.

[5] W. T. T u t t e , How to draw a graph, Proc. London Math. Soc., I l l Ser. 13 (1963), pp. 743-767.

[

] K. W a g n e r , Zwei Bemerkungen iiber Komplexe, Math. Ann. 112 (1936),

pp. 312-321.