Broersma, H.J.

Lu, M.

English

Broersma, Haitze J.

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Cycles through particular subgraphs of claw-free graphs

University of Twente Research Information

Cycles Through Particular Subgraphs of Claw-Free Graphs ~ H. J. Broersma FACULTY OF APPLIED MATHEMATICS UNIVERSITY OF TWENTE P.O. BOX217, 7500AE ENSCHEDE THE NETHERLANDS M. Lu INSTITUTE OF SYSTEMS SCIENCE ACADEMIA SlNlCA BElJlNG 100080 P.R. CHINA ABSTRACT Let G be a 2-connected claw-free graph on n vertices, and let H be a subgraph of G. We prove that G has a cycle containing all vertices of H whenever a3(H) 5 K(G), where a 3 ( H )  denotes the maximum number of vertices of H that are pairwise at  distance at  least three in G, and K(G)  denotes the connectivity of G. This result is an analog of a result from the thesis of Fournier, and generalizes the result of Zhang that G is hamiltonian if the degree sum of any K(G) + 1 pairwise nonadjacent vertices is at  least n - K(G).  0 1995 John Wiley & Sons, Inc. 1. TERMINOLOGY We use Bondy and Murty [3] for terminology and notation not defined here and consider simple graphs only. Let G be a graph of order n. The connectivity of G is denoted by K(G),  the number of vertices in a maximum independent set of G by a ( G ) ,  the set of vertices adjacent to a vertex u E V ( G )  by N ( u ) ,  and the degree of u by d ( u )  = IN(u)I. If H is a subgraph of G, we denote by a 3 ( H )  the maximum number of vertices of H that are pairwise at distance at least three in G. Let S V(G). We denote by a k ( S )  the minimum value of the degree sum (in G) of any k pairwise nonadjacent vertices of S if k 5 a(G[S] ) ,  where G [ S ]  is the subgraph of G induced by S; if k > a(G[S] ) ,  we set Journal of Graph Theory, Vol. 20, No. 4, 459-465 (1995) 0 1995 John Wiley & Sons, Inc. CCC 0364-9024/95/040459-07 460 JOURNAL OF GRAPH THEORY a k ( S )  = k ( n  - 1). Instead ofak(V(G)) we use a k ( G ) ,  and instead of a l (S)  we use the more common notation 6 ( S ) .  If H is a subgraph of G or a subset of V(G), an H-cycle of G is a cycle containing all vertices of H; a G-cycle is a Hamilton cycle, and a graph G containing a G-cycle is called hamiltonian. A graph G is claw-free if G has no induced subgraph isomorphic to K I , ~ .  2. RESULTS There are many results showing a graph G is hamiltonian if G satisfies a particular degree condition. The disadvantage of such results is that there is no result left even when only a few vertices have “small” degree, while recent results show that in these cases the graphs under consideration can still have “long” cycles containing particular sets of vertices with “large” degree. We give an example. I Theorem 1 (Dirac [5] ) .  then G is hamiltonian. let G be a graph of order n 2 3 .  If 6(G) 2 z n ,  Recently, Theorem I was generalized independently by BollobAs and Brightwell, and by Shi. Theorem 2 ([2,8]). S C V(G). If 6 ( S )  2 i n ,  then G has an S-cycle. Let G be a 2-connected graph of order n and let In a similar fashion, a generalization of the following well-known result of Chvhtal and Erdos appeared in the thesis of Fournier. Theorem 3 [4]. G is hamiltonian. Let G be a graph of order n 2 3 .  If a ( G )  5 K(G), then Theorem 4 [6]. G. If a ( H )  5 K(G), then G has an H-cycle. Let G be a 2-connected graph and let H be a subgraph of Extremal graphs for Theorems 3 and 4 can be found, e.g., within the class of complete bipartite graphs with unequal cardinalities of the two color classes. Most of these graphs contain many induced subgraphs isomorphic to K I , ~  and have a 3  = 1, while the connectivity can be arbitrarily large. The situation is quite different if we consider claw-free graphs. In Section 3 we give a proof of the following analog of Theorem 4 for claw-free graphs. Theorem 5. Let G be a 2-connected claw-free graph and let H be a subgraph of G. If a 3 ( H )  5 K(G), then G has an H-cycle. CYCLES IN CLAW-FREE GRAPHS 461 Corollary 6. and let S C V(G). If a k + , ( S )  2 n - k, then G has an S-cycle. Let G be a k-connected claw-free graph (k 2 2) on n vertices Corollary 6 is an immediate consequence of Theorem 5 and the following lemma. Lemma 7. If U ~ + ~ ( S )  2 n - k, then a3(G[S]) 5 k. Let G be a 2-connected graph on n vertices and let S C V(G) .  Prooj Suppose a?(G[S] )  L k + 1. Consider a subset T C S with (TI = k + 1 vertices that are pairwise at distance at least three in G. Then d(u)  5 n - ( k  + 1) < n  - k .  V E T  Therefore, g k + l ( S )  2 n - k implies a?(G[S] )  5 k. I Corollary 6 generalizes the following result of Zhang. Theorem 8 [9]. vertices. If ak+l (G)  2 n - k ,  then G is hamiltonian. Let G be a k-connected claw-free graph (k 2 2) on n By the way, as observed in [ l] ,  implicit in the proof of Theorem 8 in [9] is a proof of the following stronger result, which is a special case of Theorem 5 and an analog of Theorem 3. Theorem 9. then G is hamiltonian. Let G be a 2-connected claw-free graph. If a3(G)  5 K(G) ,  The proof of Theorem 5 given in Section 3 is based on proof ideas used in [ l ]  and [9]. The main idea to use so-called vertex-insertion was originally introduced in [9], and later presented more clearly in [l] .  This idea is used throughout the proof of Theorem 5 in a similar way, restricted to vertices of H.  It is clear that Theorem 5 is more general than Theorem 4 within the class of claw-free graphs. Theorem 5 is best possible in the following sense. Let G I  be a graph obtained from three disjoint complete graphs on at least three vertices by joining two disjoint triples of vertices by a triangle, where each triple contains one vertex of each complete graph. Then G I  has no hamilton cycle, is claw-free, while a 3 ( G I )  = K(GI )  + 1 = 3. We do not know whether Theorem 5 is best possible for graphs G with K ( G )  2 3. Matthews and Sumner [7] conjectured that any 4-connected claw-free graph is hamiltonian. 462 JOURNAL OF GRAPH THEORY 3. PROOF OF THEOREM 5 Let G be a 2-connected claw-free graph, let H be a subgraph of G such that cr3(H) 5 K(G),  and let C be a cycle of G containing as many vertices of H as possible, and longest subject to this condition. Fix an orientation on C. If u, u E V ( C ) ,  then u e u  denotes the consecutive vertices on C from u to u in the direction specified by the orientation of C. The same vertices, in reverse order, are given by u c u .  We will consider u c u  and U ~ U  both as paths and as vertex sets. We use u+ to denote the successor of u on C and u p  to denote its predecessor. Assuming C is no H-cycle, let ao E V ( H )  - V ( C ) .  By a variation on Menger's Theorem, there are at least K ( G )  paths connecting ao to vertices of C having only the vertex ao in common and having no internal vertices on C. Choose a maximum number of paths with this property, and denote by X = { x i  I i = 1,2, .  . . , t }  the end vertices of the paths on C,  occurring on C in the order of their indices. Obviously, 1x1 = t 1 K(G) .  First note that any pair of vertices x ; , x j  E X is joined by a path containing a0 with all internal vertices in G - V(C) .  We denote such a path by x i n - x j .  Let S j  be the set of vertices of x + C x , i ~ ,  and choose ui E Si such that u j x i  E E(G)  and lxiCu; I is as large as possible (i = 1,2, .  . . , z;  indices mod t ) .  A vertex u E Si n V ( H )  is insertible if there exist vertices u ,  u+ E V ( C )  - S j  such that u u , u u +  E E(G) ;  then uu' E E ( C )  is an insertion edge of u. We denote the set of insertion edges of u by Z(u), and we denote the first noninsertible vertex of V ( H )  occurring on u t  ex[+ by ai . Using standard techniques in hamiltonian graph theory, it is straightforward to show t + and a; exists. (3) In the rest of the proof we show that a,  and a ,  are at distance at least three in G ( i , j  E {1,2,. . . , t } ;  i # j ) ,  implying a 3 ( H )  2 t + 1 2 K ( G )  + 1, and proving Theorem 5. Suppose there exists a vertex w E u:Cx,+l fl V ( H )  n N ( x , )  and as- sume a ,  @ u:ew- .  Then every vertex of u:cw-  fl V ( H )  is insertible. Let w1 be the first vertex of u:ew f l  V ( H )  fl N ( x , )  with the property that x,-x, and x,xT are not insertion edges of the vertices in u:Cw- n V ( H )  + -, CYCLES IN CLAW-FREE GRAPHS 463 + + + (or w1 = u:). Using the cycle x ; w ~ C x , ~ x ~ C x , ~ x ~ C u ; x , ~ x ,  and inserting the vertices of u T 6 w ,  n V ( H )  in a straightforward way, we derive a contradiction. Hence If w E u:6x1;, fl V ( H )  n N(xj), then ai E u T 6 w p .  (4) By (4) and the definition of ui, we have From the definition of X and the fact that ai 6 X ,  we obtain Next we show + + There is no edge joining a vertex of u: Ca, and a vertex of u: Ca,, and I(x) n I (  y )  = 0 for all x E u:6a,  f l  V ( H )  and y E u;?a,- f l  V ( H ) .  (7) + To prove (7), suppose to the contrary there are vertices x E u:Ca, and y E uT6a,  such that xy E E(G)  or x and y have a common insertion edge uu' E E ( C )  (and x,y E V ( H ) ) .  Choose x as close to u,  along u:6u, as possible subject to the above conditions. Define LJ I ( Z )  i f x  # u: E,  = { Z E L I : ? X - ~ V . H )  0 i f x  =uT Analogously define El .  By the choice of x, we have There is no edge joining a vertex of u:6x- and a vertex of u T 6 y - ,  (8) and E; n E., = 0. (9) Consider two cases. Case 1. xy E E(G) .  By (4), there is no edge joining a vertex of u:6xp f l  V ( H )  to x,, since a ,  (Z u:Cx-. Thus E, E(C - xl). Similarly, E, C E(C - xl). By (€9, El  E(C - u 1 6 x ) .  The ver- tices of V ( C )  - (uT6x- U u : e y - )  are contained in a cycle D = + E(C - u , 6 y )  and E, 464 JOURNAL OF GRAPH THEORY c + +  + xiuiCx+x,-CyxCxJxfCujxj~xi. Clearly, Ei U E j  C E(D) .  Using (9) and similar arguments as before, we can insert the vertices of (u'Cx- U uT?y-) fl V ( H )  in D to obtain a cycle containing more vertices of H than C, a contradiction. + Case 2. Assuming Case I does not apply, choose y as close to u, along u:Ca, as possible subject to the other conditions. By the definition of insertion edges, uu' @ S, U S,. Furthermore, the choice of x and y implies x and y have a common insertion edge uu+ E E(C).  * Combining (4), (8), (9), and (lo), we get E; U E j  C E(C - ( { x i , x j }  U u i c x  U u,&)) - {uu'} and El f' E j  = 0. If u E x+?x;, define D = x i u i ~ x ; x i C ? ; u C x u + ? x : x + ? u j x j ~ x i ;  if u E xTCx;, define D = x , u j C x + x , - C u i y C u x C x J x f C u , x , ~ x i .  Clearly, in both cases E; U El C E(D), and we can derive a contradiction with the choice of C by inserting the vertices of ( u T 6 x -  U u ; e y - )  n V ( H )  in D as before. This completes the proof of (7). By ( 5 )  and (6), a ~ a ,  @ E ( G )  and N ( Q )  fl N ( a ; )  = 0, and, by (7), a ,a j  @ E(G) .  It suffices to show N ( a , )  fl N ( a , )  = 0. Suppose u E N ( a i )  fl N ( a j ) .  Then similar arguments as in the proof of Case 1 of (7) show u E V ( C ) .  By ( 5 )  and (7), u 4 u:cu; U u t e u ;  U {x j , x j } .  If u E X , ? + ~ ? S ~  U X ~ + ~ ? X ; ,  then the noninsertibility of a;  and u j ,  and (7) imply that G [ { u , u + , a i , a j } ]  = K , , 3 ,  a contradiction. If u E a'Cx;+l U ~ f C x ~ + ~ ,  assume without loss of generality, u E a+Cxi+, .  Then, considering G[{u ,  u', a , ,  a,i}] and using (7) and the noninsertibility of a j ,  we obtain aiv' E E(G) .  Let w be the first vertex of u;ca l  adjacent to u .  As before we can insert the vertices of (u ' ea ;  U u t i ' w - )  n V ( H )  in the cycle x i u ; C x ~ x 1 ~ C w ~ C a ; u + C x ~ ~ x f C u j x j ~ x ~  to derive a contradiction with the choice of C. (Note that u # x, : ,~ ,~  by (5 ) . )  Hence u E x'Cu; U xTCuj.  Assume without loss of generality u E x + 6 u j .  Considering G[{u- ,  u ,  a ; ,  aj}] and using (7) and the noninsertibility of a j ,  we obtain a;u-  E E ( G )  and u p  # xi (by ( 5 ) ) .  Choose bj E uTi 'n i  as close to uj along u f i ' u j  as possible such that ubj  E E(G) .  As before we can insert the vertices of ( u T e a ;  U uTi.0;) n V ( H )  in the cycle xiu;C u b j C x , ~ x ~ C u ~ a ; C x J x f u , C x j ~ x ,  to derive a contradiction with the choice of C. Hence the set {ao,ul, .  . . ,a ,}  consists of at least K(G) + 1 vertices of H which are pairwise at distance at least three in G, showing that a 3 ( H )  > K ( G ) ,  our final contradiction. c c  c c + +  J I +  .+ + + c t c  + + - + c + + + c CYCLES IN CLAW-FREE GRAPHS 465 References [ l ]  A. Ainouche, H. J .  Broersma, and H. J. Veldman, Remarks on hamilton- ian properties of claw-free graphs, Ars Combinat. 29C (1 990), 1 10- 12 1. [2] B. Bollobhs and G. Brightwell, Cycles through specified vertices, Combinatorica 13 (1993), 147-15s. [ 3 ]  J .A .  Bondy and U.S.R. Murty, Graph Theory with Applications Macmillan, London and Elsevier, New York (1 976). [4] V. Chv&d and P. Erdos, A note on hamiltonian circuits, Discrete Math. [ S ]  G . A .  Dirac, Some theorems on abstract graphs, Proc. London Math. [6] I. Fournier, Thkse d’Etat, LRI, Universitk de Paris-Sud, France (1985). [7] M.M. Matthews and D.P. Sumner, Hamiltonian results in KI,3-free [8] R. Shi, 2-Neighborhoods and hamiltonian conditions, J .  Graph Theory [9] C.-Q. Zhang, Hamilton cycles in claw-free graphs, J. Graph Theory 12  2 (1972), 111-113. SOC. 2 (1952), 69-81. graphs, J. Graph Theory 8 (1984), 139-146. 16 (1992), 267-271. (1988), 209-216. Received April 7, 1994 

Cycles through particular subgraphs of claw-free graphs

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