Chordality and 2-Factors in Tough Graphs

Transcription

1 Chordality and -Factors in Tough Graphs D. Bauer 1 G. Y. Katona D. Kratsch 3 H. J. Veldman 4 1 Department of Mathematical Sciences, Stevens Institute of Technology Hooken, NJ 07030, U.S.A. Mathematical Institute of the Hungarian Academy of Sciences H-1364 Budapest POB 17, Reáltanoda u , Hungary 3 Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena Jena, Germany 4 Department of Applied Mathematics, University of Twente P. O. Box 17, 7500AE Enschede, The Netherlands June 9, 1998 Astract AgraphG is chordal if it contains no chordless cycle of length at least four and is k-chordal if a longest chordless cycle in G has length at most k. Inthis note it is proved that all 3 -tough 5-chordal graphs have a -factor. This result is est possile in two ways. Examples due to Chvátal show that for all ɛ>0 there exists a ( 3 ɛ)-tough chordal graph with no -factor. Furthermore, examples due to Bauer and Schmeichel show that the result is false for 6- chordal graphs. Keywords : toughness, -factors, chordal graphs AMS Suject Classifications (1991) : 68R10, 05C38 Supported in part y NATO Collaorative Research Grant CRG Supported in part y Hungarian National Foundation for Scientific Research, OTKA Grant Numers F and T

2 1 Introduction We egin with a few definitions and some notation. Other definitions will e given later, as needed. A good reference for any undefined terms is [7]. We consider only undirected graphs with no loops or multiple edges. Let G e a graph. Then G is hamiltonian if it has a Hamilton cycle, i.e., a cycle containing all of its vertices. It is traceale if it has a path containing all of its vertices. Let ω(g) denote the numer of components of G. ThenG is t-tough if S t ω(g S) for every suset S of the vertex set V of G with ω(g S) > 1. The toughness of G, denoted τ(g), is the maximum value of t for which G is t-tough (taking τ(k n )= (n 1) for all n 1). A k-factor is a k-regular spanning sugraph. Of course, a Hamilton cycle is a -factor. We say G is chordal if it contains no chordless cycle of length at least four and is k-chordal if a longest chordless cycle in G has length at most k. Our work was motivated y a desire to understand the relationship etween the toughness of a graph and its cycle structure. For a survey of recent work in this area, see [3, 4, 5]. Toughness was introduced y Chvátal in [9]. An ovious connection etween toughness and hamiltonicity is that eing 1-tough is a necessary condition for a graph to e hamiltonian. Chvátal conjectured that there exists a finite constant t 0 such that every t 0 -tough graph is hamiltonian. This conjecture is still open. Until recently it was elieved that the smallest value of t 0 for which this might e true was t 0 =. We now know this is false. Theorem 1.1 [1]. For every ɛ>0, there exists a ( 9 ɛ)-tough nontraceale graph. 4 Chvátal also conjectured that every k-tough graph on n vertices with n k +1 and kn even has a k-factor. This was estalished in [10]. Theorem 1. [10]. LetGeak-tough graph on n vertices with n k +1and kn even. Then G has a k-factor. It was also shown in [10] that Theorem 1. is est possile. Theorem 1.3 [10]. Letk 1. For any ɛ>0, there exists a (k ɛ)-tough graph G on n vertices with n k +1and kn even which has no k-factor. The aove results imply that while -tough graphs have -factors, there exists an infinite sequence of graphs without -factors having toughness approaching. In [11] it was shown that a similar statement holds for split graphs. A graph G is called a split graph if its vertices can e partitioned into an independent set and a clique. Theorem 1.4 [11]. Every 3 -tough split graph is hamiltonian. In [[9], p.3], Chvátal found a sequence {G n } n=1 of non--factorale graphs with τ(g n ) 3. These graphs were in fact split graphs.

3 Theorem 1.5 There is a sequence {G n } n=1 τ(g n ) 3. of non--factorale split graphs with In this note we prove that all 3 -tough chordal graphs have a -factor. In fact we prove a it more. Theorem 1.6 Let G e a 3 -tough 5-chordal graph. Then G has a -factor. Since all split graphs are chordal, the graphs Chvátal constructed in [9] are also chordal. Thus Theorem 1.6 is est possile with respect to toughness. Furthermore, the graphs G l,m in [[], p.51] are 6-chordal graphs without a -factor. By choosing l and m large the toughness of these graphs can e made to approach from elow. Note that Theorem 1.6 is in some sense the definitive result of the form If G is a t-tough k-chordal graph, then G has a -factor : it follows from the examples in [9] that this is false for t< 3 and any k, y Theorem 1. it is true for t andanyk, and from the examples in [] it follows that for 3 t<theest one can hope for is a result with k =5. Unlike the case with split graphs, however, it is not true that all 3 -tough chordal graphs are hamiltonian. Theorem 1.7 [1]. For every ɛ>0 there exists a ( 7 ɛ)-tough chordal nontraceale 4 graph. Recently, Chen, Jacoson, Kézdy and Lehel [8] have shown that every 18-tough chordal graph is hamiltonian. We now conjecture the following. Conjecture: Every -tough chordal graph is hamiltonian and for every ɛ>0 there exists a ( ɛ)-tough chordal nonhamiltonian graph. Returning to -factors, it is natural to ask how large the minimum vertex degree of a t-tough (1 t<) graph can e, if the graph contains no -factor. This prolem was answered in [] for 1 t 3 and for infinitely many t satisfying 3 t<. A key lemma (Lemma 8) in [] is the asis for the proof of our main result. Of course, any paper dealing with sufficient conditions for a graph to have a regular factor relies heavily on a well-known theorem of Belck [6] and Tutte [1]. This result is given in Section. The proof of our main result appears in Section 3. 3

4 Preliminary Results Let G e a graph. If A and B are susets of V or sugraphs of G, andv V,we use e(v,b) to denote the numer of edges joining v to a vertex of B, ande(a, B) to denote v A e(v,b). We use A to denote the sugraph of G induced y A. A vertex v V will e called complete if v is adjacent to every other vertex in V, and is called simplicial if the sugraph induced y the neighorhood of v is complete. Our proof of Theorem 1.6 relies heavily on a theorem that characterizes those graphs not containing a -factor. This theorem is a special case of the theorems of Belck [6] and Tutte [1]. For disjoint susets A, B of V (G) letodd(a, B) denote the numer of components H of G (A B) withe(h, B) odd, and let Θ(A, B) = A + y B d G A (y) B odd (A, B). Theorem.1 [6], [1]. Let G e any graph. Then (i) for any disjoint sets A, B V (G), Θ(A, B) is even; (ii) the graph G does not contain a -factor if and only if Θ(A, B) for some disjoint pair of sets A, B V (G). We call a pair (A, B) of disjoint susets of V (G) withθ(a, B) atutte pair for G. Note that in any Tutte pair (A, B) forg we have B, since y definition d G A (y) odd (A, B) andsoθ(a, B) implies B > A 0. We define y B a Tutte pair (A, B) toeminimal if Θ(A, B ) 0 for any proper suset B B. Clearly any graph without a -factor contains a minimal Tutte pair. The next lemma follows easily from a result in [10]. The proof also appears in []. Lemma. Let G e a graph having no -factor. If (A, B) is a minimal Tutte pair for G, then B is an independent set. To facilitate the proof in the next section we define a Tutte pair (A, B) toea strong Tutte pair if B is an independent set. 4

5 3 Proof of Theorem 1.6 We egin with the following lemma, which is also implicit in []. Lemma 3.1 Let v e a simplicial vertex in a non-complete graph G. Then τ(g v) τ(g). Proof: First denote G v y G v. Note that if G v is complete, then τ(g v )= V (G v ) 1 V (G) = τ(g). Suppose τ(g v ) < τ(g). Then there exists X V (G v ) such that ω(g v X) and X ω(g v X) <τ(g). However ω(g X) ω(g v X), since the neighors of v in G induce a complete sugraph. But this gives X ω(g X) X <τ(g), a contradiction. ω(g v X) Proof of Theorem 1.6: Let G e a 3 -tough 5-chordal graph having no -factor and (A, B) e a strong Tutte pair for G, existing y Lemma.. Thus Θ(A, B). Let C = V (G) (A B). Since B is an independent set of vertices, d G A (y) =e(b,c). Hence y Theorem y B.1, A + e(b,c) B + odd (A, B). (1) Among all possile choices, we choose G and the strong Tutte pair (A, B) as follows: (i) V (G) is minimal; (ii) E(G) is maximal, suject to (i); (iii) B is minimal, suject to (i) and (ii); (iv) A is maximal, suject to (i), (ii) and (iii). We now show that G has properties (a)-(g) elow. (a) For any x B and any component H of C,e(x, H) 1. Proofof(a): Let x B with d G A (x) =k, andletc 1,C,...,C j denote the components of C to which x is adjacent. If j k 1, delete x from B and add x to C (thus redefining B and C). Since odd (A, B) has decreased y at most j k 1, it is easy to check that Θ(A, B) has increased y at most 1. Thus we still have Θ(A, B) (y Theorem.1(i)) and we contradict (iii). 5

6 () The vertices of A are complete. Proofof(): If not, form a new graph G y adding the edges required to make the vertices of A complete. Clearly G is still 3 -tough and (A, B) is still a strong Tutte pair for G. Oviously, no chordless cycle of G can contain a vertex of A. SinceG is 5-chordal, it follows that G is also 5-chordal. Thus we contradict (ii). (c) For any y C, e(y,b) 1. Proofof(c): Suppose that e(y,b) forsomey C. Delete y from C and add y to A (thus redefining A and C). It is easy to check that (A, B) remains a strong Tutte pair. Thus we contradict (iv). (d) Each component of C is a complete graph. Proofof(d): If not, form a new graph G y adding the edges required to make each component C 1,C,...,C s of C a complete graph. Clearly, G is still 3 -tough and (A, B) is still a strong Tutte pair for G. Assuming G is not 5-chordal, let C e a shortest chordless cycle in G of length at least 6. Clearly C can not contain a vertex of A, nor can it have more than two vertices from any component of C. SinceB is independent, C is of the form C : 1 T 1 T... k T k 1, where, for 1 i k, eacht i represents an edge t 1 i t i of a component C i in G. Form the cycle C in G y taking C and sustituting T i for T i (1 i k), where T i is a shortest t 1 i t i path in C i in G. The graph G is 5-chordal, so C has a chord. Since any chord of C must join a vertex of B and a vertex of C and C is a chordless cycle in G, we may assume, without loss of generality, that there exists a chord 1 u of C such that u is an internal vertex of some T i,sayoft m,and the cycle 1 T 1 T... m U 1,whereU is the t 1 m u supath of T m,is chordless. By (a) we have 1 <m<k.butthen 1 T 1 T... m t 1 mu 1 is a chordless cycle in G of length at least 6 which is shorter than C, contradicting the choice of C.ThusG is 5-chordal and we contradict (ii). 6

7 (e) For any y C, e(y,b)=1(andthuse(b,c)= C ). Proofof(e): Suppose now that C contains a vertex y with e(y, B) = 0. It follows from () and (d) that v is simplicial. Hence y Lemma 3.1, τ(g y) τ(g). Furthermore, (A, B) is still a strong Tutte pair for the 5-chordal graph G y. Hence, y (i), the graph G y contradicts the choice of G. (f) B. Proofof(f): We saw earlier that B > A 0, and so B 1. Suppose B = {x}. Since (A, B) is a Tutte pair with B = 1and A = 0, wehavee(b,c) odd (A, B) y (1). If e(b,c), then ω(g B) odd (A, B) e(b,c) > B, and G is not 1-tough. If e(b,c) =1,thenG is not 1-tough either. Hence B. (g) odd (A, B) =ω( C ). Proofof(g): Suppose there exists a component C i in C with e(c i,b)= C i,aneven integer. Let y e any vertex in C i.addyto A, thus redefining A and C. Itis easy to see that (A, B) is still a strong Tutte pair for G. Thus we contradict (iv). Hence G and its minimal Tutte pair (A, B) have properties (a) - (g). Set s = ω( C ) =odd(a, B). Consider the components C 1,C,...,C s of C and let y j V (C j ). Define X = A C {y 1,...,y s }. Since B is independent and e(y i,b)=1for1 i s, we have ω(g X) = B. For convenience let a = A,= B and c = C. Using properties (e), (g) and inequality (1), we have 3 X ω(g X) = a + c s = a + e(b,c) odd(a, B) a. Hence a +4. () 7

8 To complete the proof we estalish the following. Claim: c s +1. Once the claim is estalished, it follows that Thus 3 X ω(g X) = a + c s a + 1. a. (3) The fact that () and (3) are contradictory completes the argument. Proof of Claim: Form a ipartite graph F from G y deleting A and contracting each component of C into a single vertex. By (a), F has no multiple edges. The key oservation is that since G is 5-chordal, F is a forest. Otherwise, let C F e a shortest cycle in F. Then C F is of the form C F : 1 T 1 T... p T p 1, where each T i, 1 i p, represents the contracted component C i.by(d)and(e),it follows that the edges incident with each T i in C F correspond to edges i t 1 i, i+1 t i, where t 1 i t i is an edge in C i. It follows that G has a chordless cycle of length at least 6, a contradiction. Hence d F (v) =c = E(F ) V (F ) 1= + s 1. v C Thus + s 1 c and the claim is estalished. 8

9 References [1] D. Bauer, H. J. Broersma, and H. J. Veldman. Not every -tough graph is hamiltonian. Discrete Applied Math., this volume. [] D. Bauer and E. Schmeichel. Toughness, minimum degree and the existence of -factors. J. Graph Theory, 18(3):41 56, [3] D. Bauer, E. Schmeichel, and H. J. Veldman. Some recent results on long cycles in tough graphs. In Y. Alavi, G. Chartrand, O. R. Oellermann, and A. J. Schwenk, editors, Graph Theory, Cominatorics, and Applications - Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, pages John Wiley & Sons, Inc., New York, [4] D. Bauer, E. Schmeichel, and H. J. Veldman. Cycles in tough graphs - updating the last four years. In Y. Alavi and A. J. Schwenk, editors, Graph Theory, Cominatorics, and Applications - Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, pages John Wiley & Sons, Inc., New York, [5] D. Bauer, E. Schmeichel, and H. J. Veldman. Progress on tough graphs - another four years. To appear in Graph Theory, Cominatorics, and Applications - Proceedings of the Eighth Quadrennial International Conference on the Theory and Applications of Graphs. [6] H. B. Belck. Reguläre Faktoren von Graphen. J. Reine Angew. Math., 188:8 5, [7] G. Chartrand and L. Lesniak. Graphs and Digraphs. Chapman and Hall, London, [8] G. Chen, M. S. Jacoson, A. E. Kézdy, and J. Lehel. Tough enough chordal graphs are hamiltonian. Networks, 31:9 38, [9] V. Chvátal. Tough graphs and hamiltonian circuits. Discrete Math., 5:15 8, [10] H. Enomoto, B. Jackson, P. Katerinis, and A. Saito. Toughness and the existence of k-factors. J. Graph Theory, 9:87 95, [11] D. Kratsch, J. Lehel, and H. Müller. Toughness, hamiltonicity and split graphs. Discrete Math., 150:31 45, [1] W. T. Tutte. The factors of graphs. Canad. J. Math., 4:314 38,