Toughness and hamiltonicity in almost claw-free graphs

Toughness and Hamiltonicity in Almost Claw-Free Graphs -

H. J. Broersma FA CUL TY OF APPLIED M A THEMA TICS

UNIVERSITY OF TWENTE P.O. BOX 217, 7 5 0 0 A E ENSCHEDE

THE NETHERLANDS

Z. Ryjatek DEPARTMENT OF MATHEMATICS UNIVERSITY OF WEST BOHEMIA

306 14 PLZEN CZECH REP UBLIC

I. Schiermeyer LEHRSTUHL c FUR MATHEMATIK

TECHNISCHE HOCHSCHULE AACHEN 0-52056 AACHEN

GERMANY

ABSTRACT

Some known results on claw-free (Kl,3-free) graphs are generalized to the larger class of almost claw-free graphs which were introduced by RyjaEek. In particular, we show that a 2-connected almost claw-free graph is I-tough, and that a 2-connected almost claw-free graph on n vertices is hamiltonian if 6 2 i ( n - 2), thereby (partly) generalizing results of Matthews and Sumner. Finally, we use a result of Bauer et al. to show that a 2-connected almost claw-free graph on n vertices is hamiltonian if d(u) + d(u) + d(w) 2 n for all independent sets of vertices u, u , and w. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

We use Bondy and Murty [3] for terminology and notation not defined here and consider simple graphs only.

Throughout, let G be a graph of order 11. 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

Journal of Graph Theory, Vol. 21, No. 4, 431-439 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/04043 1-09

432 JOURNAL OF GRAPH THEORY

a vertex u by N ( u ) , and the degree of u by d(u ) = IN(.)[ . We denote by c+k(G) the minimum value of the degree-sum of any k pairwise nonadjacent vertices if k 5 a ( G ) ; if k > a ( G ) , we put ak(G) = k ( n - 1). Instead of (TI(G) we use the more common notation 6(G) . If G has a Hamilton cycle (a cycle containing every vertex of G ) , then G is called hamiltonian; G is called hamiltonian-connected if every two vertices of G are connected by a Hamilton path (a path containing every vertex of G). The graph G is t-tough ( t E R, t L 0) if IS1 2 t * w(G - S ) for every subset S of V ( G ) with w(G - S) > 1, where w(G - S) denotes the number of components of G - S. The toughness of G , denoted T ( G ) , is the maximum value of t for which G is t-tough (T(K,) = a for all n 2 I ) . A dominating set of G is a subset S of V ( G ) such that every vertex of G belongs to S or is adjacent to a vertex of S. The domination number, denoted y ( G ) , is the minimum cardinality of a dominating set of G . The graph G is claw-free if G has no induced subgraph isomorphic to K I , ~ . A vertex u E V ( G ) is a center of a claw if u has three pairwise nonadjacent neighbors. If H is a subgraph of G and S is a subset of V ( G ) or a subgraph of G , then N H ( S ) denotes the set of all vertices of H having a neighbor in S ; if S = {u}, we write N ( u , G ) for the subgraph induced by N G ( { u } ) . We define the local independence number ~ L ( G ) and the local domination number ~ L ( G ) as follows:

Obviously, y ( G ) 5 a ( G ) and yL(G) 5 aL(G) for every graph G . Moreover, it is easy to see that G is claw-free if and only if ~ L ( G ) 5 2 .

Following RyjkEek [ I I], we say a graph G is almost claw-free if there exists an independent set A C V ( G ) such that a ( N ( u , G ) ) 2 2 for every u # A and y ( N ( v , G ) ) 5 2 < a ( N ( u , G ) ) for every u E A. Equivalently, G is almost claw-free if ~ L ( G ) 5 2 and the set A consisting of the centers of all claws is an independent set. Clearly, every claw-free graph is almost claw- free, and there exist almost claw-free graphs which are not claw-free. In [ I 11 it was shown that every almost claw-free graph has no induced subgraph isomorphic to K1,5 or K1,1,3.

Our objective is to generalize results on claw-free graphs to almost claw-free graphs. In Section 2 we prove that in a noncomplete almost claw-free graph G , T ( G ) 2 m i n ( 1 , ~ K(G)} , thereby (partly) generalizing a result of Matthews and Sumner [8]. In Section 4 we prove that a 2-connected almost claw-free graph G is hamiltonian if S(G) 2 7 ( n - 2). This result generalizes another result of Matthews and Sumner [9]. Finally, we use a result of Bauer et al. [ I ] to show that a 2-connected almost claw-free graph G is hamiltonian if a3(G) 2 n .

1

I

2. TOUGHNESS

1 Let G be a noncomplete graph. Then it is obvious that T ( G ) 5 7 K ( G ) . If G is claw-free, then equality holds, as was shown by Matthews and Sumner.

Theorem 1 [8].

In the same paper they conjecture that every 4-connected claw-free graph is hamiltonian. This conjecture is a special case of the following well-known conjecture due to Chvktal.

Conjecture 2 [ 5 ] .

I If G is a noncomplete claw-free graph, then T ( G ) = 7 K ( G ) .

Every 2-tough graph on n 2 3 vertices is hamiltonian.

TOUGHNESS AND HAMILTONICITY 433

Even in the case of claw-free graphs, a possible proof of the conjecture seems to be very difficult. Before we discuss some results on hamiltonicity involving degree conditions in Section 3, we first prove the following result which generalizes Theorem 1 in case K(G) 5 2.

Theorem 3.

Proof.

I If G is a noncomplete almost claw-free graph, then T ( G ) 2 min(1, 3 K(G)} .

In any noncomplete graph G, clearly T(G) 5 3 K(G) . If G is not connected, then T ( G ) = 5 K(G) = 0. Suppose G # K , is a connected almost claw-free graph and S is a cutset of G such that T ( G ) = IS l / (w(G - S)) < min(1, K(G)} . Let H I , . . . , H , be the components of G - S. There exist at least K(G) disjoint paths from u E V ( H , ) to u E V ( H j ) for any i , j E (1,. . . , p } with i # j . Each of these paths contains a vertex of S. Hence for each i E (1 , . . . , p } there are at least K ( G ) edges joining vertices of H , to distinct vertices of S. Thus there are at least ~ K ( G ) edges from C - S to S, counting at most one from any component of G - S to a particular vertex of S. Suppose every vertex u E S has neighbors in at most two components of G - S. Then there are at most 2)SI edges from G - S to S, counting at most one from any component of G - S to a particular vertex of S. Then ~ K ( G ) 5 2)SI or 7 K ( G ) 5 I S ) / p = T ( G ) , a contradiction.

Hence S contains a center x of a claw with neighbors in at least three components of G - S. Since G is almost claw-free, r ( N ( x , G ) ) 5 2. This implies that there exists a neighbor y of x in S, and, moreover, that x has neighbors in at least three components of G - S, and y is adjacent to vertices in precisely two of these components. But then T = S - { y } is a cutset of G with w ( G - T ) = w ( G - S) - 1 , so that T ( G ) 5 ITl/(w(G - T ) ) = (1.S;' - l ) / ( w ( G - S) - 1) < ISl / (w(G - S)) = T ( G ) , a contradiction. Hence T ( G ) 2 min(1, 3 K(G)} .

The graph Go of Figure 1 shows that we cannot prove an analogue of Theorem 1 for almost claw-free graphs with connectivity exceeding two. Go is almost claw-free (the set of centers of claws is A = (s2,s3}) and 3-connected, but if we let S = (SI,S~,S~,S~}, then IS1 = 4, w(G - S) = 3 and hence

I

I

I

I

FIGURE 1. A 3-connected almost claw-free graph Go which is not :-tough.


3. HAMILTON CYCLES

There are many results showing that claw-free graphs have interesting hamiltonian properties under certain additional assumptions. Here we focus on degree conditions ensuring hamiltonicity. The following result is due to Matthews and Sumner.

Theorem 4 [9]. hamiltonian.

I If G is a 2-connected claw-free graph with S(G) 2 3 ( n - 2), then G is

The following generalization of Theorem 4 was independently obtained by Broersma and Zhang.

Theorem 5 [4,13]. hamiltonian.

If G is a 2-connected claw-free graph with a3(G) 2 n - 2, then G is

More generally, Zhang [13] proved that G is hamiltonian if G is a k-connected claw-free graph with ~ rk+ l (G) 2 n - k(k 2 2).

Theorem 4 was extended to classes of graphs containing a restricted number of claws by Flandrin and Li [7].

An analogue of Theorem 5 for K I , 1,3-free and K2.3-free graphs was obtained by Flandrin, Jung, and Li [6].

In Section 4 we prove the following two results; the first generalizes Theorem 4, the second is an analogue of Theorem 5. These results are independent of the aforementioned results of Flandrin and Li, and Flandrin, Jung, and Li, respectively.

Theorem 6. is hamiltoni an.

Theorem 7. hamiltonian.

Theorem 6 is best possible. but we do not know whether Theorem 7 is best possible. Perhaps Theorem 5 can be generalized to almost claw-free graphs.

The examples we know showing that Theorem 6 is best possible are the same examples that show Theorem 4 to be best possible, and they all have connectivity 2. It is likely that the degree bound in Theorem 6 can be improved for 3-connected graphs, as it is the case with Theorem 4 (as shown by Zhang's result). To show that Theorem 6 is more general than Theorem 4, consider the following graphs, one of which is drawn in Figure 2.

Let HI , H2, H3 be three vertex disjoint copies of K s ( S 2 2) and join two new vertices x and y to all vertices of H2 and H3. Join x also to y and to all vertices of HI. Let G be a graph obtained from this graph by adding k 2 1 edges such that NHI(H2) = 0, N H ~ ( H ~ ) f' N H ~ ( H ~ ) = 0 and N,jI(H2) U NHI(H3) f V(H1). (For an example with S = 3, k = 3, N H , ( H ~ ) = { a } and N H , ( H ~ ) = {h}, see Figure 2.) Then G is a 2-connected almost claw-free graph with n = 36 + 2 vertices and hence G satisfies the assumptions of Theorem 6, but it does not satisfy the assumptions of Theorem 4 since it is not claw-free. (Note that G also does not satisfy the assumptions of the main result of [7].)

I If G is a 2-connected almost claw-free graph with S(G) 2 3 (n - 2), then G

If G is a 2-connected almost claw-free graph with cq(G) 2 n , then G is

4. PROOFS OF THEOREMS 6 AND 7

We first introduce some additional notation and prove two auxiliary results.


H2 H3

FIGURE 2. A graph satisfying the assumptions of Theorem 6 but not of Theorem 4

Let C be a cycle of G. By 6 we denote the cycle C with a given orientation, and b y 6 the same cycle with the reversed orientation. If u , u E V ( C ) , then u 6 u denotes the consecutive vertices on C from u to u in the direction specified by 6. The same vertices, in reverse order, are given by u t u . We will consider U ~ V and uku 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.

Lemma 8. Let be a longest cycle in an almost claw-free graph G . Let y E V ( G ) - V ( C ) and let x be a neighbor of y on C such that x-x' @ E ( G ) . Then there exists a vertex d E V ( C ) n N ( x - ) f l N ( x ) n N ( x + ) with the following properties: Either d' = x- or d- = x', or there is a path Ql between d- and d' and a path QZ between x - ( x ' ) and x such that V ( Q 1 ) fl V(Q2) = 0 and V ( Q 1 ) U V ( Q 2 ) = { x - , x , x + , d - , d , d ' } .

Suppose first that y and x - have a common neighbor u in N ( x ) . It is clear that the choice of C implies u E V ( C ) , and y u - , yu' @ E(G). Since G is almost claw-free and x is a center of a claw, u is not a center of a claw, implying that u-u' E E(G) . We can extend C by replacing u-uu' by u - u + , and x-x by x - u y x , a contradiction. Hence y and x- have no common neighbor in N ( x ) . By symmetry, y and x + have no common neighbor in N ( x ) . Since yL(G) 5 2, there is a vertex d E N ( x ) dominating both x- and x' . It is obvious that d E V ( C ) and that d is not a center of a claw. If d+ = x- or d- = x+, then we are done. Suppose d+ # x - and d- # x'. Consider the subgraph of G induced by {d - ,d , d',x'}. At least one of the edges d-d ' , d-x' and d'x' belongs to G . If d-d+ E E ( G ) , then put Q , = d-d' and Qz = x-dx'x. If d-x' E E ( G ) , then put Ql = d-x'dd' and Qz = x-x. If d+x+ E E(C) , then put Ql = d-dx'd+ and Q2 = x -x .

Proof.

The similar statement for x+ follows by symmetry.

In the sequel, let G be a nonhamiltonian 2-connected almost claw-free graph, let

I be a

longest cycle in G, and let H be a component of G - V ( C ) . Denote by X I , . . . ,xk the vertices


of N c ( H ) occurring on i. in the order of their indices, and let Si = xT?xi1 and si = ISil. Clearly, k 2 2. Let l i denote the length of a longest path between xi and xi+[ with all internal vertices in H ( i = 1 , . . . , k ; indices mod k ) . Note that in the proof of Lemma 9 we sometimes apply Lemma 8 to 6.

Proof Let i E { 1,. . . , k } and let Li denote a path of length Ei between xi and xi+i with all internal vertices in H (indices mod k ) . If we compare the length of C with the length of the cycle obtained from C by replacing xiS;xi+l by x iLix i+l , we obtain that si 2 l i - 1 by the choice of C ( i = 1 , . . . , k ) . We can however sometimes increase this lower bound on si by looking more carefully at the “configuration” concerning the vertices of Si and its neighbors x i and xi+l. If, e.g., xlTxT E E ( C ) , then we can increase the lower bound on si by 1 by observing that instead of replacing x;Six i+l in C by xiLixi+l, we can replace xL7x;Sixi+l in C by ~ ~ T x + x ~ L ~ x ~ + ~ . In this case we say that the “left gain” on si is 1. Similarly, if xl~+lx:+l E E ( G ) , we can increase the lower bound on s; by 1. In this case we say that the “right gain” on s; is 1. Note that in this case the left and right gain are additive, i.e., we can increase the lower bound on s; by 2 if both xix; E E ( G ) and xI<lx:+l E E(G) .

More generally, we define the lefl gain gL(si) on si as the amount we can add to the lower bound l i - 1 on si by only looking at the configuration concerning x i (to be specified later), and the right gain gR(sj) on si as the amount we can add to this lower bound by only looking at the configuration concerning xi+l ( i = 1 , . . . , k ; indices mod k ) .

We will show that the left and right gains on si we obtain in the sequel are additive with one exception which needs a more careful analysis.

We first obtain values for g R ( S i ) and gL(si+l) by looking at the configuration concerning x;+l ( i = 1,. . . , k ; indices mod k ) . For this purpose we distinguish the following possible configurations concerning x i + I.

A. x i + l x ~ i E E(G) . We already showed that in this case g ~ ( s i ) = 1 and gL(si+l) = 1. B. xl;ix;1 @! E(G). Then xi+l is a center of a claw, and there exists a vertex d E

V ( C ) associated to xi+l with the properties given by Lemma 8. The cycles

dCx;+i show that d @ { X ~ ~ ~ , X ~ + ~ , X ~ ~ ~ } , respectively. (Note that in the second case d = xi+2 is not the center of a claw, and hence xi+2xT+2 E E(G).)

B1. d E Si+ l . Then gR(si) = 0 and g ~ ( s i + i ) = d,-(d,xi+l) 2 2, since otherwise replacing XlqIXi+iSi+lXi+2 in C by xl<IdCxi+lLi+lxi+2 we obtain a longer cycle. Here &(u, u ) denotes the distance along C (i.e., the distance in the subgraph of G induced by E ( C ) ) between two vertices of C.

B2. d E Si. By similar arguments as in B1, we obtain gL(si+l) = 0 and gR(s i ) =

B3. In the other cases, gL(si+l) = 1; otherwise, (using the terminology of Lemma 8) replacing d-dd’ by Q l and x1Glxi+~Si+lx;+2 by ~ i , ~ Q 2 x i + l L ; + l x i + 2 we obtain a longer cycle. Similarly, g R ( s i ) = 1.

c + ’ + - - + - - ~ ‘ + I ~ C ~ ; + I L ~ + I X ~ + ~ C X ~ + , , xL’+~dLi+ 1x i+ iCx i+2~i+2C~i+1> and xi+iLi+lxi+2cxi+l

c

dc(d ,x i+l ) 2 2.

As we argued before, gL(s i ) and gR(s i ) are additive if xlrxT E E ( G ) and E E(G). It is not difficult to check that the same is true if only one of those edges is present, or in case gL(si) = 0 or gR(si) = 0. We can however not always guarantee the additivity in case


x l r x + , xI<lx:+1 @ E ( G ) and g ~ ( s ; ) , g R ( s ; ) > 0. Then x; andx;+I are (nonadjacent) centers of a claw, and there are vertices d l , d2 E V ( C ) associated to xi and x ;+ I , respectively, with the properties given by Lemma 8. As before, it is clear that dl , dz @ { x i , xi, x:, xi+ I, xi+ 1 , x:+ I}. We give a more detailed analysis of the possible cases.

Suppose first dl = dz. Since dl is not a center of a claw, at least one of dlxlT and d l x : is an edge of G. Then, however, the cycles d lex ,d l kx ,+ lL;x ;ex i+ ld l and d lex iL ;x ;+lkx ; dl Cx:+ldl, respectively, contradict the choice of C. Hence dl # d2. Suppose next dld2 E E ( C ) . If d~ = d:, then the cycle d I ~ x ; L ; x ; + I e d 2 x l ~ I E x , t d I contradicts the choice of C ; if dl = dz-, then the cycle d l ~ x ; + I L ; x ; E d z x i + l ~ x + d l contradicts the choice of C . Hence dld2 @ E(C) . Using the properties given in Lemma 8, the above observations yield that g L ( s ; ) and g R ( S j ) are additive in this case.

If d2 @ x ; e d l , then g L ( s ; ) and g R ( s ; ) are again additive. Suppose now d2 E x ; e d l . We first show that dl @ (d2,d:). If dl = d2, then the following cycles, respectively, show that did: @ E ( G ) and x i d : @ E(G) : x ; L ; x ; + l ~ d : d l ~ x + d l x i + , I e x ; and xFdTex[+ldl kx;Lix,+lex, . Then, since dl is not a center of a-claw, we obtain x i d l E E(G) , and, by symmetry, x:+,d: E E(G) . Now the cycle x l ~ d I C x , L ; x , + ~ d I n ; + I C d I x;+ lCx; contracts the choice of C. If dl = d:, then the cycle x i d ~ ~ x , + ~ d z C x i L , x ; + ~ ~ x , contradicts the choice of C . From these observations we conclude that g L ( s i ) = dc(x i ,d l ) 2 4 and g R ( s i ) = dc(xi+1, d2) 2 4. Clearly, g L ( s ; ) and ~ R ( s ; ) are not additive in this case, but we could regard the total gain on s; as two additive gains of at least 2 at each side of S; . In such cases we redefine g L ( s i ) = 2 and g R ( s i ) = 2.

If we adapt the definition of g L ( s ; ) and g ~ ( s ; ) in the way we discussed above (in Case 2), we obtain the following conclusion.

+ Cuse 1. dl E x;+~Cx;.

+ - -

+ Cuse 2. dl E x ; C x i + l .

- - + + + -

1 Proof of Theorem 6. Assume 6(G) 2 ?; (n - 2). Using Lemma 9, we obtain n 2

IfZl si + k + 1 2 I i + 2k + 1 2 4k + 1 2 9. Suppose V ( H ) = {u} . Then 7 ( n - 2) 5 6(G) 5 d ( u ) 5 k 5 4 ( n - l ) , a contradiction. Hence no component of G - V ( C ) is an isolated vertex. We may assume IV(H)I 2 2. Among the pairs U I , u2 E V ( H ) for which

1

I

INc(ul) l + INc(u2)I is as large as possible (1)

choose a pair u, u such that

IN,-(u) U Nc(u)l is as large as possible. (2)

If INc(u) U Nc(u)l 5 1, then (1) and (2) imply INc(H)I 5 1, a contradiction. Hence INc(u) U Nc(u)l 2 2. Moreover, by the 2-connectedness of G, we may assume u and u are chosen in such a way that uy l , uy2 E E ( G ) for two distinct vertices y1, y2 E V(C). Let p =

INc(u)l, q = INc(u)l, r = INc(u) f' Nc(u)l. Assume p 2 q without loss of generality, and


let l ( u , u ) denote the length of a longest path between u and u in H . Denote N c ( u ) U N c ( u ) by {XI, . . . , x,}, where the vertices occur on C in the order of their indices. Then, using Lemma 9 for this subset {XI,. . . ,x,} of N c ( H ) , we obtain

+

n 2 IV(H)I + IV(C)l 2 IV(H)I + 1% + t i = l

I

L IV(H)I + 1 li + 2t 2 IV(H)I + 4t + max(2,r) . I ( u , u ) . (3) i = l

We distinguish two cases.

By the choice of u and u , d H ( u I ) + dH(u2) 2 26(G) - ( p + q ) 2 6(C) + 1 2 5 ( n + 1) for all u I , u2 E V(H). By Theorem 3, G is I-tough. Using a result of Bauer and Schmeichel [2], and Tian and Zhao [12], IV(C)l 2 26(G) + 2, hence IV(H)I 5 n - (26(G) + 2) 5

~ ( n - 2). Thus d H ( u I ) + dH(u2) 2 IV(H)I + 1 for all V I , u2 E V(H), implying H is hamiltonian-connected by a result of Ore [lo]. In particular, l ( u , u ) = IV(H)I - 1. Using

Case I. p + q 5 S(G) - 1.

I

(3), we have

Clearly,

Hence

n 2 IV(H)I + 4t + 2l(u,u) 2 31V(H)I + 4t

S(G) + 1 - q 5 IV(H)I.

n L 36(G) + 4t - 3q + 1 = 36(G) + t + 3(t .

2 36(C) + 3 2 n + 1,

2 .

9 ) + 1

(4)

a contradiction. p + q 2 S(G).

Using (3) and (4), we have Case 2.

n L IV(H)I + 4t + max(2,r) . l ( u , u )

2 S(G) + 1 - q + 4 ( p + q - r ) + max(2, r } . I(u, u )

= S(G) + 1 + 2 ( p + q) + ( p + q - r ) + p - 3r + max(2,r) . I (u ,u)

L 36(G) + 3 + p - 3r + max(2,r) . I (u ,u )

I n + 1 + max(2,r) . I (u ,u) - 2 r .

This clearly yields a contradiction in case l ( u , u ) 2 min(2, r } . For the remaining cases assume I ( u , u ) = 1 and r 2 I (u ,u) + 1 . Then N H ( u ) f l N H ( u ) = 0, hence IV(H>I 2 26(C) - ( p + 4) . BY (31,

n 2 IV(H)I + 4t + r

2 2S(G) - ( p + q ) + 4 ( p + q - r ) + r

= 2S(G) + ( p + q ) + ( p + q - r ) + ( p + q - 2r )

2 36(G) + 2 2 n .


This implies p = q = r = 2, 6 ( G ) = 4, n = 14 and IV(H)I = 4. Now u and u have neighbors w! and w2 in H , respectively, such that w ! wz, u w l , uw2 @ E ( G ) (since l (u , u ) = 1). Furthermore, d~(w1) + d~(w2) = 2 since IV(H)I = 4, while on the other hand the choice of u and u implies d H ( w I ) + d~(w2) 2 26(G) - ( p + q ) = 8 - 4 = 4, a contradiction.

Assume ( T ~ ( G ) 2 n. By Theorem 3, G is 1-tough. We use the following lemma. The first part of this lemma is [ 1, Theorem 51 and the second part is implicit in the proof of [ 1 , Theorem 91.

Lemma. Let G be a 1-tough graph on n 2 3 vertices with u3(G) 2 n. Then every longest cycle of G is a dominating cycle. Moreover, if G is nonhamiltonian, G contains a longest cycle C such that max{d(u)lu E V ( G ) - V ( C ) } 2

Let C be a dominating cycle such that there is a vertex u E V ( G ) - V ( C ) with d ( u ) 2 I k L 3 a3(G) 2 f n. By Lemma 9 (with d ( u ) = k), n 2 I,=, s, + k + 1 2 X I = , 1, + 2k + 1 2 4k + 1 2 3 n + 1 , a contradiction.

Note added in proofi true with n replaced by n - 2 for n 2 79.

I Proof of Theorem 7.

q ( G ) .

4 I

H. Li and F. Tian recently proved a result implying that Theorem 7 is

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[3] J. A. Bondy and U. S. R. Murty, Graph Theoty with Applications, Macmillan, London

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Received April 4, 1995

Toughness and hamiltonicity in almost claw-free graphs

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