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Toughness and the Cycle Structure of Graphs
Quo Vadis, Graph Theory? J. Gimbel, J.W. Kennedy & L.V. Quintas (eds.) A m a h of Discrele Marhematics, 55, 145-152 (1993) 0 1993 Elsevier Science Publishers B.V. All rights reserved.
TOUGHNESS AND THE CYCLE STRUCTURE OF GRAPHS
Douglas BAUER Department of Pure and Applied Mathematics Stevens Institute of Technology, Hoboken, New Jersey, U.S.A.
Edward SCHMEICHEL Department of Mathematics and Computer Science San Jose State University, San Jose, California, U.S.A.
Abstract We discuss some old and new problems concerning the relationship between the toughness of a graph and its cycle structure.
1.
Introduction
Since ChvAtal introduced the notion of toughness in [l] significant progress has been made toward understanding the relationship between this parameter and the cycle structure of a graph. Much of this progress is surveyed in [2]. However many vexing problems remain. Some of these problems were raised in [l] but others are relatively new. The purpose of this note is to discuss recent progress in this area and indicate some directions for future research. Before proceeding further we present a few definitions and some notation. Additional definitions will be given later as needed. A good reference for any undefined terms is [3]. We consider only finite undirected graphs without loops or multiple edges. Let w(G) denote the number of components of a graph G. A graph G is t-tough if IS12 to(G - S) for every subset S of the vertex set Vof G with o(G - 4 > 1. The toughness of G , denoted t(G), is the maximum value o f t for which G is t- tough (t(K,) = = for all n 2 1). We let a(G) denote the cardinality of a maximum set of independent vertices of G. The length of a longest cycle in G is called the circumference of G and is denoted c(G). We also letp( G) denote the length of a longest path in G. A k-factor is a k-regular spanning subgraph. For k 22, we let
ok = min {
k
I
d ( v i ) {vl, v2, ..., vk } is an independent set cf vertices } .
i= 1
2. A Direction for Future Research Does there exist a constant to such that every to-tough graph is Hamiltonian? In [ l ] ChvAtal conjectured that such a to does exist and noted that to = 2 would imply a theorem of Fleischner [4], stating that the square of every 2-connected graph is Hamiltonian. ChvAtal also conjectured that every 3/2-tough graph has a 2-factor and that every k-tough graph on n vertices with kn even has a k-factor. Only the latter conjecture is correct, as shown by Enomoto et al. [5] in Theorems 1 and 2 below.
Theorem 1: Let G be a k-tough graph on n vertices with ti 2 k +1 and kn even. Then G has a k-factor.
D. Bauer and E. Schmeichel
146
Theorem 2: Let k 11.For any positive real number E, there exists a ( k - &)-toughgraph G on n vertices with kn even and n 2 k + 1 which has no k-factor. Since we are interested in the cycle structure of graphs we focus for now on k = 2. Theorems 1 and 2 state, in essence, that 2-tough graphs have 2-factors and that there exist (2 - E)tough graphs without 2-factors. The infinite family of graphs in [5] that demonstrate the latter all have vertices of degree 4. Can such graphs be found with minimum degree at least 5? What if we require 6 ( G )2 alV(G)I for some constant a > O? More generally, we raise the following questions. Let G be a t-tough graph on n vertices, where 1 5 t 12. Find the smallest nonnegative constants P ( t ) and X t ) such that for sufficiently large n (1)
6(G) 2 P( t)n implies G is Hamiltonian
(2) 6 ( G )2 y(t)n implies G has a 2-factor. Clearly p(t) 2y(t)and if the conjecture that 2-tough graphs are Hamiltonian is correct then P(2) =$2) = 0. We first outline recent progress on question (2). By Theorem 1, y(2) = 0. The following theorem is established in
[a.
Theorem 3:
2-2 For 1 I t < 2 let G be a t-tough graph on n 2 3 vertices. If 6(G) 2 ( -) n, then G contains a l+t 2-factor.
It is demonstrated in [6] that Theorem 3 is best possible for 1 I t <3/2; consequently y ( t ) = 2 - t / 1 + t in that range. Furthermore, if 3 /2 S t < 2, then $ 2 ) Y ( t ) = (7t ) andin fact y(t) = f i t ) for every t of the form (2r - 1) / r , where r 22. The diff&&e betweLn At) and (2 - t)/(1 + t) is quite small since .94 I(t 2 - 1)/(7t- 7 - t2)I1 for 2 E [3/2,2]. It is also shown in [6] that if 312 I t < 2, then y(t) 2g(t), where g(t) is defined as follows. Let r = r(t) 2 2 denote )(2-1) and the integer such that (2r - l ) / r I t I (2r + l ) / ( r+ 1) and set g, ( t ) = (r+1)(1+1)-5 1-1 . Then
(e) '_;!
i
g 2 ( t ) = 3 ( r + l ) ( 2 1 - 3 ) +1+1
g(t) =
g l ( t ) if ( 2 r - I ) / r I t < b ( r ) g2(t) if b ( r ) l t < ( 2 r + l ) / ( r + 3 )
where b(r) = (6r2- r - 4)l(3r2+ r - 3 ) Note that g ( t ) is continuous at t = b ( r ) and that g(t) =f(t) if t = (2r - l ) / r for r 1 2. For other values of t the difference between f(t) and g(t) is quite small, e.g., g( 1I n ) = 3/19 = .I58 and f(l1/7) = 4/25 = .16. We conjecture that in fact r(t) = g(t) if 312 I t < 2.
In contrast to our knowledge of 'I(t),we know almost nothing about p(t). We do know that p(1) = 1/2since the following theorem of Jung [7] is best possible. Theorem 4: Let G be a 1-tough graph on n 2 11 vertices with 0 2 2 n - 4. Then G is Hamiltonian.
In [I%] it is shown that there exists an infinite collection of non-Hamiltonian 1-tough graphs on n vertices with 6 2 = n - 5 and thus p(1) = 112. If we assume t(G) > 1 the bound on o2in
Toughness and the cycle structure of graphs
147
Theorem 4 can be lowered, but not by very much [9]
Theorem 5: Let G be a graph on n 230 vertices with t(G)> 1. If
02
2 n - 7, then G is Hamiltonian.
In [9] it is also shown that there exists an infinite collection of non-Hamiltonian graphs whose toughness is larger than 1 and with 0, = n - 8. Recently, the non-Hamiltonian 1-tough graphs for which 6 3 2 (3n - 24)12 have been characterized [lo]. Theorem 5 also follows from this characterization. Let us now assume that t is a fixed rational number such that 1 < t 12. For such t determining p(t) is an open problem. We can obtain a (possibly crude) upper bound on p ( t ) from a result in [ll], given below.
Theorem 6: Let G be a 1-tough graph on n2 3 vertices with 032 n. Then c(G) 2min (n,n + 03 I 3 -a). Clearly 01 I n/(t + 1) and 03 2 36 and so the next corollary follows.
Corollary 7: For 1 1 t 1 2 let G be a t-tough graph on n vertices with 6 2 n / ( t+ 1). Then G is Hamiltonian. Thus we conclude that p(t) I 1/(t + 1) for 1 I t 1 2 . We now suggest the intriguing possibility that y(t) = p(t) for 1 I t 1 2 . Since Theorem 4 is also best possible for the existence of 2factors, we know y(1) = p( 1) = 112. If the conjecture that 2-tough graphs are Hamiltonian is correct then y(2) = p(2) = 0. However for an intermediate value of t, say t =3/2, all we know is that 1/5 = y(3/2) I p(3I2) _<2/5.Clearly there is much work to be done. Some evidence that $ 2 ) = p ( t ) for 1 1 t 1 2 is that a similar phenomenon occurs with respect to the binding number. For v E V ( G )let N(v) denote the set of vertices in G which are adjacent to v, and for S V(G)let N ( S ) = ,N (v) . The binding number of G, denoted b(G),is the minimum of IN(S)I/ISI taken over all nonempty subsets S of V(G)such thatN(S) # V(G).In [I21 Woodall proved that if b(G) 2312 then G is Hamiltonian. He also showed that the constant 3/2 is best possible, both for the existence of a Hamiltonian cycle as well as for the existence of a 2-factor. While this might lead one to believe that y(t) = p(t) it is important to realize that binding number and toughness have different characteristics. For instance it is NP-hard to determine if a graph G is t-tough for any fixed rational number t [13], while b( G ) can be computed in polynomial time [14]. 3.
More Open Problems
Another question regarding the relationship between toughness and cycle structure concerns what we will refer to as the Dominating Cycle Property (DCP). A cycle C in a graph G is a dominating cycle if every edge of G has at least one of its vertices on C. A graph G has the DCP if every longest cycle in G is a dominating cycle. Dominating cycles were introduced by and discussed extensively in a later paper by Veldman [16]. Recently, Nash-Williams in the DCP has proved to be a useful idea in the study of cycle structure in graphs (e.g., [2], [l 13, [17]-[19]). The relationship between toughness and the DCP is not known for t > 1. For t = 1 we have the following results, each of which is best possible. The first theorem is a result due
[la
D. Bauer and E. Schmeichel
148
to Bigalke and Jung [8]. Theorem 8:
Let C be a 1-tough graph on n vertices with 6 2n13. Then G has the DCP. This was later generalized in [ll]. Theorem 9:
Let G be a 1-tough graph on n vertices with 03 2n. Then G has the DCP. Since every Hamiltonian graph has the DCP it is natural to add a third question to the two raised in the previous section. Let G be a t-tough graph on n vertices, where 1 5 t 52. Find the smallest nonnegative constant q(f) such that for sufficiently large n (3) 6 ( G )2 q ( f ) nimplies G has the DCP.
0
Clearly q(r)Ikt)and if the conjecture that 2-tough graphs are Hamiltonian is true, q(2) = p(2) = 0. But how does q(f) compare with $t)? Since Theorem 8 is best possible, 113 = q ( l ) < s 1 ) = 112. Also since q(t)and are nonincreasing functions of rand $514) = 113, we have q(t)ssr)for 1It I514. Is it the case that q(t) 5 $t) for 514 < t I 2 ?
sr)
Another interesting problem is to find c(G),given t( G) and 6(@. Here we know very little, even for 1-tough graphs. The following theorem appears in [20]. Theorem 10:
Let G be a 1-tough graph on n I 3 vertices with 6 2 n13. Then c(G)2min ( n , n + 6 - a + 1).
Corollary 11: Let G be a 1-tough graph on n 2 3 vertices with6 2 n13. Then c(G)2 5n16 + 1. However, we do not believe that Corollary 11 is best possible. In fact we conjecture [ 111 that under the hypothesis in Corollary 11, c ( G )2 ( l l n + 3)112. We have stated our questions in terms of 6 rather than bk ( k 22). Of course they can be considered in terms of 0 k and in fact Theorem 10 has recently been generalized in this direction [21], as shown below (compare with Theorem 6 ) . Theorem 12: 0
Let G be a I-tough graph on n 2 3 vertices with 0 3 2 n . Then c(G)2 min (n, n+ 2 - a + 1). 3 With regard to cycle structure problems it is often difficult to generalize a theorem involving a lower bound on 6to a similar theorem involving a lower bound on ok.Surprisingly this is not the case with respect to $t). With little additional effort we established the following in
[6J.
Theorem 13:
Let G be a t-tough graph on n 2 3 vertices with 1 5r < 2. If
Toughness and the cycle structure of graphs
149
then G has a 2-factor. Another open problem concerns the relationship between the toughness of a graph and whether the graph is pancyclic. A graph G on n vertices ispuncyclic if G contains a cycle of length 1 for every 1 such that 3 I 1 In.In [ 11 ChvAtal conjectured that there exists a constant to such that every to-tough graph is pancyclic. Of course this question is still open. An easier (and hence more frustrating) question was recently raised by Jackson and Katerinis [22]; namely, does there exist a constant to such that every t,-tough graph has a triangle? A problem that has received some recent interest concerns the growth of c(G) as a function of n for fixed t. More specifically, let %(t, n) denote the class of all 2-connected t-tough graphs on n vertices and let q t , n) = min {c(G)I G E G2(t, n)}. As n +=,will q t , n) + for fixed t? The answer is yes, although if t is not fixed and G2(t, n) is replaced by the class of all k-connected graphs, the answer is no; e.g., C(Kk,n - k ) = 2k for all n 2 k. The following appears in [B]. 00
Theorem 14:
For a fixed constant A depending only on t, C(t, n) log C(t,n) 2 A log n. Conjecture 15:
For a fixed constant A depending only on t, C(t, n) 2 A log n. It is shown in [23] that Conjecture 15 is true for 3-connected graphs. Additional evidence for the conjecture is that a similar result holds for paths. Let G,(t, n) denote the class of all connected t-tough graphs on n vertices and let P(t,n) =min {dG)I G E G,(t, n)}. Theorem 16:
For a fixed constant A depending only on t, P(t,n) 2 A log n. It is shown in [23] that Conjecture 15, if true, and Theorem 16 are essentially best possible f o r t S 1. It is an open problem to determine best possible lower bounds on q t , n) and P(t, n) f o r t > 1. We conclude with some comments on the relationship between toughness, minimum degree, and existence of k-factors. The following result [24] generalizes both Theorem 1 and Theorem 3. Theorem 17:
Let G be a t-tough graph on n vertices and k 2 2 an integer such that n 1k + 1 and kn is even. If 6(G) 2 ( k - 1)( k - t) n/(1 + t) + ( k - 2), then G contains a k-factor. Theorem 17 is meaningful only if t > k- 1 - Ilk. It remains an open problem to determine if the degree bound in Theorem 17 is best possible in this range. If t > (2k2 - 2k - 1)/(2k - l), Theorem 17 strengthens a result proved by Katerinis [25] and independently by Egawa and Enomoto [261.
D. Bauer and E. Schmeichel
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Theorem 18: Let G be a graph on n vertices and k 2 1 an integer such that n 2 4k - 5 and kn is even. If 6(G) 2 n I2 then G contains a k-factor.
Acknowledgements Douglas Bauer was supported in part by the National Security Agency under Grant MDA 904-H-89-2008. Edward Schmiechel was supported in part by the National Science Foundation under Grant DMSS904520.
References V. Chvhtal; Tough graphs and Hamiltonian circuits, Discrete Math., 5,215-228 (1973). D. Bauer, E.F. Schmeichel, and H.J. Veldman; Some recent results on long cycles in tough graphs, Proc. 6th Int. Conf. on the Theory and Applications ojGraphs, Kalamam, 1988, Y. Alavi, G. Chartrand, O.R. Oellermaun and A.J. Schwenk (editors), 113-123 (1991). G. chartrand and L. Lesniak; Graphs and Digraphs, Wadsworth,Inc., Belmont, calif.(1986). H. Fleischner; The square of every 2-connected graph is Hamiltonian, J. Cornbinatorial Theory Ser. 8. 16.29-34 (1974). H. Enomoto, B. Jackson, P. Katerinis, and A. Saito; Toughness and the existence of k-factors, J. Graph Theory, 9.87-95 (1985). D. Bauer and E.F. Schmeichel; Toughness, minimum degree and the existence of 2-factors, preprint (1991). H.A. Jung; On maximal circuits in finite graphs, Annals ojDiscrele Math., 3, 129-144 (1978). A. Bigalke and H.A. Jung; Uber Hamiltonische Kreise and unabhangige Ecken in Graphen, Monatsh. Math., 88, 195-210 (1979). D. Bauer, G. Chen, and L. Lasser;A degree condition for Hamiltonian cycles in t-tough graphs with t >1, preprint (1991). H.A. Jung, Shwe Kyaw, and Wei Bing; private communication. D. Bauer, A. Morgana, E.F. Schmeichel, and H.J. Veldman; Long cycles in graphs with large degree sums, Discrete Math., 79,SP-70 (1989/90). D.R. Woodall; The binding number of a graph and its Anderson number, J. Cornbinatorial Theory Ser. B , 2 9 , 2 7 4 6 (1973). D. Bauer, S.L. Hakimi, and E.F. Schmeichel; Recognizing tough graphs is NP-hard, Discrete Appl. Math.,28, 191-195 (1990). W.H. Cunningham; Computing the binding number of a graph, Discrete Appl. Math., 27, 283-285 (1M). C. St. J.A. Nash-Williams; Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency, Studies in Pure Mathematics, Academic Press, London, 157-183 (1971). H.J. Veldman; Existence of dominating cycles and paths, Discrete Math., 43,281-2% (1983). D. Bauer, G. Fan, and H.J. Veldman; Hamiltonian properties of graphs with large neighborhood unions, Discrete Math., 9 6 , 3 3 4 9 (1991). D. Bauer, H.J. Broersma, and H.J. Veldman; Around three lemmas in Hamiltonian graph theory, Topics in Cornbinutorics and Graph Theory, Physica-Verlag,Heidelberg, 101-1 10 (1990). Bert Fassbender; A sufficient condition on degree sums of independent triples for Hamiltonian cycles in I-tough graphs, preprint (1989). D. Bauer, E.F. Schmeichel, and H.J. Veldman; A generalization of a theorem of Bigalke and Jung, Ars Cornbinatoria, 26.53-58 (1988). Vu-Dinh-Hoa;Note on a theorem of Bauer, Morgana, Veldman and Schmeichel, preprint (1990). B. Jackson and P. Katerinis; A characterizationof 312-tough cubic graphs, preprint (1990). H.J. Broersma, J. van den Heuvel, H.J. Veldman, and H.A. Jung; Long paths and cycles in tough graphs, prepI.int (1991). D. Bauer and E.F. Schmeichel; Toughness, minimum degree, and the existence of k-factors, preprint (1991).
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[25] P. Katerinis; Minimum degree of a graph and the existence of k-factors. Proc. Indian Acad. Sci. (Math. [26]
Sci.), 94, 123-127 (1985). Y. Egawa and H. Enomoto; Sufficient conditions for the existence of k-factors. Recent Studies in Graph Theory,V.R. Kulli (editor), Vishwa International Publications, 96-105 (1989).
TOUGHNESS AND THE CYCLE STRUCTURE OF GRAPHS
Douglas BAUER Department of Pure and Applied Mathematics Stevens Institute of Technology, Hoboken, New Jersey, U.S.A.
Edward SCHMEICHEL Department of Mathematics and Computer Science San Jose State University, San Jose, California, U.S.A.
Abstract We discuss some old and new problems concerning the relationship between the toughness of a graph and its cycle structure.
1.
Introduction
Since ChvAtal introduced the notion of toughness in [l] significant progress has been made toward understanding the relationship between this parameter and the cycle structure of a graph. Much of this progress is surveyed in [2]. However many vexing problems remain. Some of these problems were raised in [l] but others are relatively new. The purpose of this note is to discuss recent progress in this area and indicate some directions for future research. Before proceeding further we present a few definitions and some notation. Additional definitions will be given later as needed. A good reference for any undefined terms is [3]. We consider only finite undirected graphs without loops or multiple edges. Let w(G) denote the number of components of a graph G. A graph G is t-tough if IS12 to(G - S) for every subset S of the vertex set Vof G with o(G - 4 > 1. The toughness of G , denoted t(G), is the maximum value o f t for which G is t- tough (t(K,) = = for all n 2 1). We let a(G) denote the cardinality of a maximum set of independent vertices of G. The length of a longest cycle in G is called the circumference of G and is denoted c(G). We also letp( G) denote the length of a longest path in G. A k-factor is a k-regular spanning subgraph. For k 22, we let
ok = min {
k
I
d ( v i ) {vl, v2, ..., vk } is an independent set cf vertices } .
i= 1
2. A Direction for Future Research Does there exist a constant to such that every to-tough graph is Hamiltonian? In [ l ] ChvAtal conjectured that such a to does exist and noted that to = 2 would imply a theorem of Fleischner [4], stating that the square of every 2-connected graph is Hamiltonian. ChvAtal also conjectured that every 3/2-tough graph has a 2-factor and that every k-tough graph on n vertices with kn even has a k-factor. Only the latter conjecture is correct, as shown by Enomoto et al. [5] in Theorems 1 and 2 below.
Theorem 1: Let G be a k-tough graph on n vertices with ti 2 k +1 and kn even. Then G has a k-factor.
D. Bauer and E. Schmeichel
146
Theorem 2: Let k 11.For any positive real number E, there exists a ( k - &)-toughgraph G on n vertices with kn even and n 2 k + 1 which has no k-factor. Since we are interested in the cycle structure of graphs we focus for now on k = 2. Theorems 1 and 2 state, in essence, that 2-tough graphs have 2-factors and that there exist (2 - E)tough graphs without 2-factors. The infinite family of graphs in [5] that demonstrate the latter all have vertices of degree 4. Can such graphs be found with minimum degree at least 5? What if we require 6 ( G )2 alV(G)I for some constant a > O? More generally, we raise the following questions. Let G be a t-tough graph on n vertices, where 1 5 t 12. Find the smallest nonnegative constants P ( t ) and X t ) such that for sufficiently large n (1)
6(G) 2 P( t)n implies G is Hamiltonian
(2) 6 ( G )2 y(t)n implies G has a 2-factor. Clearly p(t) 2y(t)and if the conjecture that 2-tough graphs are Hamiltonian is correct then P(2) =$2) = 0. We first outline recent progress on question (2). By Theorem 1, y(2) = 0. The following theorem is established in
[a.
Theorem 3:
2-2 For 1 I t < 2 let G be a t-tough graph on n 2 3 vertices. If 6(G) 2 ( -) n, then G contains a l+t 2-factor.
It is demonstrated in [6] that Theorem 3 is best possible for 1 I t <3/2; consequently y ( t ) = 2 - t / 1 + t in that range. Furthermore, if 3 /2 S t < 2, then $ 2 ) Y ( t ) = (7t ) andin fact y(t) = f i t ) for every t of the form (2r - 1) / r , where r 22. The diff&&e betweLn At) and (2 - t)/(1 + t) is quite small since .94 I(t 2 - 1)/(7t- 7 - t2)I1 for 2 E [3/2,2]. It is also shown in [6] that if 312 I t < 2, then y(t) 2g(t), where g(t) is defined as follows. Let r = r(t) 2 2 denote )(2-1) and the integer such that (2r - l ) / r I t I (2r + l ) / ( r+ 1) and set g, ( t ) = (r+1)(1+1)-5 1-1 . Then
(e) '_;!
i
g 2 ( t ) = 3 ( r + l ) ( 2 1 - 3 ) +1+1
g(t) =
g l ( t ) if ( 2 r - I ) / r I t < b ( r ) g2(t) if b ( r ) l t < ( 2 r + l ) / ( r + 3 )
where b(r) = (6r2- r - 4)l(3r2+ r - 3 ) Note that g ( t ) is continuous at t = b ( r ) and that g(t) =f(t) if t = (2r - l ) / r for r 1 2. For other values of t the difference between f(t) and g(t) is quite small, e.g., g( 1I n ) = 3/19 = .I58 and f(l1/7) = 4/25 = .16. We conjecture that in fact r(t) = g(t) if 312 I t < 2.
In contrast to our knowledge of 'I(t),we know almost nothing about p(t). We do know that p(1) = 1/2since the following theorem of Jung [7] is best possible. Theorem 4: Let G be a 1-tough graph on n 2 11 vertices with 0 2 2 n - 4. Then G is Hamiltonian.
In [I%] it is shown that there exists an infinite collection of non-Hamiltonian 1-tough graphs on n vertices with 6 2 = n - 5 and thus p(1) = 112. If we assume t(G) > 1 the bound on o2in
Toughness and the cycle structure of graphs
147
Theorem 4 can be lowered, but not by very much [9]
Theorem 5: Let G be a graph on n 230 vertices with t(G)> 1. If
02
2 n - 7, then G is Hamiltonian.
In [9] it is also shown that there exists an infinite collection of non-Hamiltonian graphs whose toughness is larger than 1 and with 0, = n - 8. Recently, the non-Hamiltonian 1-tough graphs for which 6 3 2 (3n - 24)12 have been characterized [lo]. Theorem 5 also follows from this characterization. Let us now assume that t is a fixed rational number such that 1 < t 12. For such t determining p(t) is an open problem. We can obtain a (possibly crude) upper bound on p ( t ) from a result in [ll], given below.
Theorem 6: Let G be a 1-tough graph on n2 3 vertices with 032 n. Then c(G) 2min (n,n + 03 I 3 -a). Clearly 01 I n/(t + 1) and 03 2 36 and so the next corollary follows.
Corollary 7: For 1 1 t 1 2 let G be a t-tough graph on n vertices with 6 2 n / ( t+ 1). Then G is Hamiltonian. Thus we conclude that p(t) I 1/(t + 1) for 1 I t 1 2 . We now suggest the intriguing possibility that y(t) = p(t) for 1 I t 1 2 . Since Theorem 4 is also best possible for the existence of 2factors, we know y(1) = p( 1) = 112. If the conjecture that 2-tough graphs are Hamiltonian is correct then y(2) = p(2) = 0. However for an intermediate value of t, say t =3/2, all we know is that 1/5 = y(3/2) I p(3I2) _<2/5.Clearly there is much work to be done. Some evidence that $ 2 ) = p ( t ) for 1 1 t 1 2 is that a similar phenomenon occurs with respect to the binding number. For v E V ( G )let N(v) denote the set of vertices in G which are adjacent to v, and for S V(G)let N ( S ) = ,N (v) . The binding number of G, denoted b(G),is the minimum of IN(S)I/ISI taken over all nonempty subsets S of V(G)such thatN(S) # V(G).In [I21 Woodall proved that if b(G) 2312 then G is Hamiltonian. He also showed that the constant 3/2 is best possible, both for the existence of a Hamiltonian cycle as well as for the existence of a 2-factor. While this might lead one to believe that y(t) = p(t) it is important to realize that binding number and toughness have different characteristics. For instance it is NP-hard to determine if a graph G is t-tough for any fixed rational number t [13], while b( G ) can be computed in polynomial time [14]. 3.
More Open Problems
Another question regarding the relationship between toughness and cycle structure concerns what we will refer to as the Dominating Cycle Property (DCP). A cycle C in a graph G is a dominating cycle if every edge of G has at least one of its vertices on C. A graph G has the DCP if every longest cycle in G is a dominating cycle. Dominating cycles were introduced by and discussed extensively in a later paper by Veldman [16]. Recently, Nash-Williams in the DCP has proved to be a useful idea in the study of cycle structure in graphs (e.g., [2], [l 13, [17]-[19]). The relationship between toughness and the DCP is not known for t > 1. For t = 1 we have the following results, each of which is best possible. The first theorem is a result due
[la
D. Bauer and E. Schmeichel
148
to Bigalke and Jung [8]. Theorem 8:
Let C be a 1-tough graph on n vertices with 6 2n13. Then G has the DCP. This was later generalized in [ll]. Theorem 9:
Let G be a 1-tough graph on n vertices with 03 2n. Then G has the DCP. Since every Hamiltonian graph has the DCP it is natural to add a third question to the two raised in the previous section. Let G be a t-tough graph on n vertices, where 1 5 t 52. Find the smallest nonnegative constant q(f) such that for sufficiently large n (3) 6 ( G )2 q ( f ) nimplies G has the DCP.
0
Clearly q(r)Ikt)and if the conjecture that 2-tough graphs are Hamiltonian is true, q(2) = p(2) = 0. But how does q(f) compare with $t)? Since Theorem 8 is best possible, 113 = q ( l ) < s 1 ) = 112. Also since q(t)and are nonincreasing functions of rand $514) = 113, we have q(t)ssr)for 1It I514. Is it the case that q(t) 5 $t) for 514 < t I 2 ?
sr)
Another interesting problem is to find c(G),given t( G) and 6(@. Here we know very little, even for 1-tough graphs. The following theorem appears in [20]. Theorem 10:
Let G be a 1-tough graph on n I 3 vertices with 6 2 n13. Then c(G)2min ( n , n + 6 - a + 1).
Corollary 11: Let G be a 1-tough graph on n 2 3 vertices with6 2 n13. Then c(G)2 5n16 + 1. However, we do not believe that Corollary 11 is best possible. In fact we conjecture [ 111 that under the hypothesis in Corollary 11, c ( G )2 ( l l n + 3)112. We have stated our questions in terms of 6 rather than bk ( k 22). Of course they can be considered in terms of 0 k and in fact Theorem 10 has recently been generalized in this direction [21], as shown below (compare with Theorem 6 ) . Theorem 12: 0
Let G be a I-tough graph on n 2 3 vertices with 0 3 2 n . Then c(G)2 min (n, n+ 2 - a + 1). 3 With regard to cycle structure problems it is often difficult to generalize a theorem involving a lower bound on 6to a similar theorem involving a lower bound on ok.Surprisingly this is not the case with respect to $t). With little additional effort we established the following in
[6J.
Theorem 13:
Let G be a t-tough graph on n 2 3 vertices with 1 5r < 2. If
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then G has a 2-factor. Another open problem concerns the relationship between the toughness of a graph and whether the graph is pancyclic. A graph G on n vertices ispuncyclic if G contains a cycle of length 1 for every 1 such that 3 I 1 In.In [ 11 ChvAtal conjectured that there exists a constant to such that every to-tough graph is pancyclic. Of course this question is still open. An easier (and hence more frustrating) question was recently raised by Jackson and Katerinis [22]; namely, does there exist a constant to such that every t,-tough graph has a triangle? A problem that has received some recent interest concerns the growth of c(G) as a function of n for fixed t. More specifically, let %(t, n) denote the class of all 2-connected t-tough graphs on n vertices and let q t , n) = min {c(G)I G E G2(t, n)}. As n +=,will q t , n) + for fixed t? The answer is yes, although if t is not fixed and G2(t, n) is replaced by the class of all k-connected graphs, the answer is no; e.g., C(Kk,n - k ) = 2k for all n 2 k. The following appears in [B]. 00
Theorem 14:
For a fixed constant A depending only on t, C(t, n) log C(t,n) 2 A log n. Conjecture 15:
For a fixed constant A depending only on t, C(t, n) 2 A log n. It is shown in [23] that Conjecture 15 is true for 3-connected graphs. Additional evidence for the conjecture is that a similar result holds for paths. Let G,(t, n) denote the class of all connected t-tough graphs on n vertices and let P(t,n) =min {dG)I G E G,(t, n)}. Theorem 16:
For a fixed constant A depending only on t, P(t,n) 2 A log n. It is shown in [23] that Conjecture 15, if true, and Theorem 16 are essentially best possible f o r t S 1. It is an open problem to determine best possible lower bounds on q t , n) and P(t, n) f o r t > 1. We conclude with some comments on the relationship between toughness, minimum degree, and existence of k-factors. The following result [24] generalizes both Theorem 1 and Theorem 3. Theorem 17:
Let G be a t-tough graph on n vertices and k 2 2 an integer such that n 1k + 1 and kn is even. If 6(G) 2 ( k - 1)( k - t) n/(1 + t) + ( k - 2), then G contains a k-factor. Theorem 17 is meaningful only if t > k- 1 - Ilk. It remains an open problem to determine if the degree bound in Theorem 17 is best possible in this range. If t > (2k2 - 2k - 1)/(2k - l), Theorem 17 strengthens a result proved by Katerinis [25] and independently by Egawa and Enomoto [261.
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Theorem 18: Let G be a graph on n vertices and k 2 1 an integer such that n 2 4k - 5 and kn is even. If 6(G) 2 n I2 then G contains a k-factor.
Acknowledgements Douglas Bauer was supported in part by the National Security Agency under Grant MDA 904-H-89-2008. Edward Schmiechel was supported in part by the National Science Foundation under Grant DMSS904520.
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