Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z)  n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

Bauer, D.

Schmeichel, E.

Veldman, H.J.

English

AbstractLet G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) ⩾ n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

Elsevier - Publisher Connector 

A note on dominating cycles in 2-connected graphs 

Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

NARCIS 

A note on dominating cycles in 2-connected graphs

University of Twente Research Information

DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 155 (1996) 13-18 A note on dominating cycles in 2-connected graphs D. B a u e r  a,,,1, E. S c h m e i c h e l  b,2, H.J.  V e l d m a n  c a Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ 07030, USA b Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192, USA e Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands Received 10 December 1992 Abstract Let G be a 2-connected graph on n vertices such that d(x)+ d(y)+ d(z)>/n for all triples of independent vertices x,y,z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs. 1. Introduction We strengthen a result in [6] concerning dominating cycles in 1-tough graphs. First we give a few definitions and some notation. We consider only finite undirected graphs with no loops or multiple edges. A good reference for any undefined terms is [15]. Let o~(G) denote the number of components of  a graph G. A graph G is t-tough if  IS[ >~tco(G - S )  for every subset S of the vertex set V(G) with o g (G -  S) > 1. The toughness of G, denoted t(G), is the maximum value of t for which G is t-tough (taking t(K,) = e~ for all n~>l). We let ct(G) denote the maximum cardinality of a set of  independent vertices of G, and c(G) be the length of a longest cycle in G. A cycle C of G is called a dominating cycle if  every edge of G has at least one of its vertices on C. We let d(v) denote the degree of vertex v. For k<<,ct(G) we let ak(G) = min{~-]ki=ld(vi)l{Vl ..... vk} is an independent set of k vertices } and NCk(G) be the minimum cardinality of the neighborhood union of any k such vertices. If G is a graph on n vertices, then for k > ct(G) we set * Corresponding author. 1 Supported in part by the National Security Agency under Grant MDA 904-89-H-2008. 2 Supported in part by the National Science Foundation under Grant DMS 9206991. 0012-365X/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSD1 0 0 1 2 - 3 6 5 X ( 9 4 ) 0 0 3 6 4 - 5  14 D. Bauer et al./Discrete Mathematics 155 (1996) 13-18 as(G) = k(n - ~(G)) and NCk(G) = n - ~(G). If G has a noncomplete component we let NC2(G) denote the minimum neighborhood union of any pair of vertices at distance two apart; otherwise NC2(G) = n - 1. If no ambiguity can arise we sometimes write instead of ~(G), etc. There are now many results in graph theory relating the toughness of a graph to its cycle structure [9,10]. However recently it was discovered [4] that for any fixed positive rational number t, it is NP-hard to recognize t-tough graphs. Nevertheless, toughness remains an important parameter in the study of hamiltonian cycles in graphs. In particular, the truth of Chv~ital's well-known conjecture [16] that 2-tough graphs are hamiltonian would have a number of useful applications [1]. On the other hand, the fact that it is NP-hard to recognize 1-tough graphs makes it desirable to strengthen some theorems by replacing the assumption that a graph G is 1-tough with the weaker assumption that G is 2-connected. The idea is to try to draw the same conclusion regarding the cycle structure of G under the weaker hypothesis of 2-connectedness by specifying an easily described family of exceptional graphs for which the new theorem does not hold. To illustrate this type of improvement consider the following theorem, established independently by Bondy [12], Bermond [11], and Linial [18]. Theorem 1. Let G be a 2-connected 9raph on n vertices. Then c(G)>~ min(n, a2). An analogous theorem for 1-tough graphs was later established by Bauer and Schme- ichel [8] and independently by Tian and Zhao [20]. Theorem 2. Let G be a 1-touoh 9raph on n>.3 vertices. Then c (G )~  min(n, a2 +2).  Both of these theorems were generalized in the manner described above by the following result, presented three years ago at this conference [5]. We call a graph H a az-subgraph of a 2-connected graph G if H is a 2-connected spanning subgraph of G with a2(H) = a2(G). Define the graph Sh,p,q to be the graph obtained from ({u} + pKh) tO ({v} + qKh) by adding the edge uv. Let Th,p,q = K1 + Sh,p,q. Theorem 3. Let G be a 2-connected nonhamiltonian 9raph on n vertices with c(G) <~ a2 + 1. Then G is a a2-subgraph o f  one of  the followin9: (1) K~ + tKl, where t > s >~ 3; (2) Ks + (K2 to tK1), where t>>.s>~3; ( 3 )  K2 a t- (gl , t  to qK2), where q>>.2,t>>.3; (4) K3 + qK2, where q>~4; (5) K2 + tKh, where t >>. 3, h >>. l ; (6) K2 +(tKh UKh_1), where t>~2,h>>.2; (7) K2 + (tKh tA Kh+l), where t>~2,h>>. 1; (8) Th,p,q, where p>>.2,q>>.2,h>>.l. D. Bauer et al. / Discrete Mathematics 155 (1996) 13-18 15 As another illustration, Skupiefi [19] was able to characterize the nonhamiltonian 2- connected graphs with a21> n -  4, thereby improving the following well-known theorem of Jung [17]. Theorem 4. Let G be a 1-tough graph on n >~ 11 vertices with o-2 >>.n- 4. Then G is hamiltonian. The purpose of this note is to provide a similar improvement to the following result, established in [6]. Theorem 5. Let G be a 1-tough graph on n vertices with o-3 >~n. Then every longest cycle in G is a dominating cycle. An analogous theorem for 2-connected graphs had already been established by Bondy [131. Theorem 6. Let G be a 2-connected graph on n vertices with o-3 >>- n + 2. Then every longest cycle in G is a dominating cycle. Our main result, proved in Section 2, generalizes the previous two theorems. Consider the following four classes of graphs: (1) 1£2 + (Ka U Kb U Kc ) where a,b,c >~ 2; (2) K3 + (aK2 U bK3), where a,b>~O and a + b = 4; (3) Ks + (sK2 U K3), where s ~> 4; (4) K s + ( s +  1)K2, where s>~4. Let ~ i  be the class of all spanning subgraphs of graphs in class (i), 1 ~<i~<4, and let ~ = U/4=lJct°i. Theorem 7. Let G be a 2-connected graph on n vertices with o-3 >~n and suppose G ~ ~ .  Then every longest cycle in G is a dominating cycle. A number of recent results in hamiltonian graph theory rely on the knowledge that every longest cycle in a graph G is a dominating cycle [2,9,10]. Many of these theorems employ either Theorem 5 or Theorem 6, i.e., they require that either G is 1-tough with o-3 >t n or that G is 2-connected with o-3 >/n + 2. As a consequence of Theorem 7, it is no longer necessary to check the NP-hard hypothesis of Theorem 5. 2. Proof of Theorem 7 To prove Theorem 7 we make use of a stronger version of Theorem 5. In [6] it was noted that the condition that G be 1-tough in Theorem 5 can be replaced by a weaker condition. Let ~ol(G) denote the number of components of G having order at least 16 D. Bauer et al./ Discrete Mathematics 155 (1996) 13-18 two. A graph G is called c o l -  tough if o91(G - S) lSI for every nonempty proper subset S of  the vertex set V(G).  The proof of the following result is implicit in the proof of  Theorem 5 of [6]. Theorem 8. Let  G be an Ogl-tOugh graph on n vertices with 63 ~ n. Then every long- est cycle in G is a dominating cycle. The assumption that G is ~Ol-tOugh in Theorem 8 is a natural one since a necessary condition for a graph to have a dominating cycle is that it be col-tough [21]. Theorem 8 generalizes Theorem 6 and also yields the following simple proof of  Theorem 7. Proof of Theorem 7. Suppose G satisfies the hypothesis of the theorem and contains a longest cycle that is not a dominating cycle. Then by Theorem 8 there exists a nonempty set of  vertices S such that G - S contains at least IS[ + 1 nontrivial com- ponents. Let s = ISI. Since G is 2-connected, s~>2. Let G1,G2 . . . . .  Gs+l+j (j>~0) be the nontrivial components of  G -  S, and let n i  : IV(Gi)I, 1 <~i<<.s q. 1 q . j .  Let t be the number of  trivial components of  G - S. Assume, without loss of  generality, that 2 ~< nl ~<n2 ~< .- .  ~<ns+l+j. I f  s : 2, then G E 9(F1 (noting that our characterization is in terms of spanning subgraphs). Suppose s >/3. Clearly n >>. n l +n2 q.n3 q," • "q.ns+l+j q-s+ t and n<~a3 <<.nl q.n2 q,n3 - 3 + 3 s - m i n ( 1 , t ) .  Hence 2(sq . j -2)<. . .naq . . . .+ns+l+j<~2s  - t - min(1,t) - 3, implying that 2j~< 1 - t - min(1,t). We conclude that j = t -- 0 and n~+l E {2,3}. Hence i f s  = 3, G E ~ 2 .  Finally, if s~>4, then ns = 2, so G C ~(Y3 if n~+l = 3 and G E o~4 if  ns+l = 2. [] 3. Applications Let us now return to Theorem 5. The following strengthening of  this theorem is also in [6]. Theorem 9. Let  G be a 1-tough graph on n vertices with aa~>n~>3. Then every longest cycle in G is a dominating cycle. Moreover, i f  G is nonhamiltonian, G contains a longest cycle C such that max{d(v) lv  ~ V ( G ) -  V(C)}>/tr3/3. The second part of  Theorem 9 is implicit in the proof of  [6, Theorem 9]. The details of  the argument can be found in the appendix of [7]. Using techniques analogous to those used to prove Theorem 7 we can also strengthen Theorem 9 so that its hypothesis can be checked in polynomial time. Theorem 10. Let  G be a 2-connected nonhamiltonian graph on n vertices with tr3 >~n and suppose G ([ 9f ~. Then G contains a longest cycle C (necessarily dominating) and a vertex Vo E V ( G ) -  V(C)  such that d(vo)>>-a3/3 unless G is a spanning subgraph o f  a graph in one o f  the following six  classes o f  graphs. D. Bauer et al./Discrete Mathematics 155 (1996) 13-18 17 ( 1 ) K2 + (Kl U K~ U Kb), where a > b >1 1; (2) K3 + (Kl U 2K3 U Ka), where 1 <<. a <~ 3; (3) Ks +(K1UK3 U ( s -  1)K2), where s>~3," (4) Ks + (2K1 UK3 tO ( s -  2)K2), where s>>.3," (5) Ks + (K1 U sK2), where s>>.3; (6) Ks + ( 2 K 1 U ( s -  1)K2), where s>~3. As a consequence of Theorems 7 and 10 a number of recent results can be improved in the manner previously described. Three of them are given below. Theorem 11 (Baner et al. [6]). Let  G be a 1-tough graph on n vertices with o-3 >~n>~ 3. Then c(G)>>, min(n,n + o3/3 - ~). Theorem 12 (Baner er al. [3]). Let  G be a 1-tough graph on n vertices with o-3 >>.n >~ 3. Then c( G ) >~ min(n, 2NC2). Theorem 13 (Broersma et al. [14]).. Let  G be a 1-tough graph on n vertices with a3 >~ n q- r >~ n >~ 3 and n >>. 8k - 6r - 17. Then c( G ) >~ min( n, 2NCk ). References [1] D. Bauer, H.J. Broersma, J. van den Heuvel and H.J. Veldman, On hamiltonian properties of 2-tough graphs, J. Graph Theory 18 (1994) 539-543. [2] D. Bauer, H.J. Broersma and H.J. Veldman, Around three lemmas in hamiltonian graph theory, in: R. Bodendiek and R. Henn, ed., Topics in Combinatorics and Graph Theory (Physica - Verlag, Heidelberg, 1990) 101-110. [3] D. Bauer, G. Fan and H.J. Veldman, Hamiltonian properties of graphs with large neighborhood unions, Discrete Math. 96 (1991) 33-49. [4] D. Bauer, S.L. Hakimi and E. Schmeichel, Recognizing tough graphs is NP-hard, Discrete Appl. Math. 28 (1990) 191-195. [5] D. Bauer, H.A. Jung and E. Schmeichel, On 2-connected graphs with small circumference, J. Combin. Inform. System Sci. 15 (1990) 16-24. [6] D. Bauer, A. Morgana, E. Schmeichel and H.J. Veldman, Long cycles in graphs with large degree sums, Discrete Math. 79 (1989/90) 59-70. [7] D. Bauer, A. Morgana, E. Schmeichel and H.J. Veldman, Long cycles in graphs with large degree sums, Memorandum No. 618, Faculty of Applied Mathematics, University of Twente, Enschede, The Netherlands, 1987. [8] D. Bauer and E. Schmeichel, Long cycles in tough graphs, Stevens Research Reports in Mathematics 8612, Stevens Institute of Technology, Hoboken, N J, 1986. [9] D. Bauer, E. Schmeichel and H.J. Veldman. Some recent results on long cycles in tough graphs, in: Y. Alavi et al., ed., Graph Theory, Combinatorics, and Applications (Western Michigan University, Kalamazoo, MI, 1991) 113-123. [10] D. Bauer, E. Schmeichel and H.J. Veldman, Cycles in tough graphs - updating the last 4 years, Proc. of the Seventh Intemat. Conf. on Graph Theory, Combinatorics, Algorithms, and Applications, (Western Michigan University, Kalamazoo, MI, 1992), to appear. [11] J.C. Bermond, On hamiltonian walks, in: C. St. J. A. Nash-Williams and J. Sheehan, eds., Proc. Fifth British Combinatorial Conference (Congressus Numerantium XV, Utilitas Mathematica, Winnipeg, 1976) 41-51. [12] J.A. Bondy, Large cycles in graphs, Discrete Math. 1 (1971) 121-131. 18 D. Bauer et al./Discrete Mathematics 155 (1996) 13-18 [13] J.A. Bondy, Longest paths and cycles in graphs of high degree, Technical Report CORR 80-16, University of Waterloo, Waterloo, Ontario, 1980. [14] H.J. Broersma, J. van den Heuvel and H.J. Veldman, Long cycles, degree sums and neighborhood unions, Discrete Math. 121 (1993) 25-35. [15] G. Chartrand and L. Lesniak, Graphs and Digraphs. (Wadsworth, Belmont, 1986). [16] V. Chvfital, Tough graphs and hamiltonian circuits, Discrete Math. 5 (1973) 215-228. [17] H.A. Jung, On maximal circuits in finite graphs, Ann. Discrete Math. 3 (1978) 129-144. [18] N. Liniat, A lower bound on the circumference of a graph, Discrete Math. 15 (1976) 297-300. [19]Z. Skupiefi, An improvement of Jung's condition for hamiltonicity, 30. Internationales Wissenschaitliches Kolloquium, TH llmenau (GDR) 5 (1985) 111-113. [20] Y. Tian and L. Zhao, On the circumference of 1-tough graphs, Kexue Tongbao 35 (1988) 538-541. [21] H.J. Veldman, Existence of dominating cycles and paths, Discrete Math. 43 (1983) 281-296. 

DISCRETE 
MATHEMATICS 
ELSEVIER Discrete Mathematics 155 (1996) 13-18 
A note on dominating cycles in 2-connected 
graphs 
D. Bauer  a,,,1, E. Schmeiche l  b,2, H.J. Ve ldman c 
a Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ 07030, USA 
b Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192, 
USA 
e Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, 
Netherlands 
Received 10 December 1992 
Abstract 
Let G be a 2-connected graph on n vertices uch that d(x)+ d(y)+ d(z)>/n for all triples 
of independent vertices x,y,z. We prove that every longest cycle in G is a dominating cycle 
unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of 
graphs. 
1. Introduction 
We strengthen a result in [6] concerning dominating cycles in 1-tough graphs. First 
we give a few definitions and some notation. We consider only finite undirected graphs 
with no loops or multiple edges. A good reference for any undefined terms is [15]. 
Let o~(G) denote the number of components of a graph G. A graph G is t-tough if 
IS[ >~tco(G -S )  for every subset S of the vertex set V(G) with og(G-  S) > 1. The 
toughness of G, denoted t(G), is the maximum value of t for which G is t-tough 
(taking t(K,) = e~ for all n~>l). We let ct(G) denote the maximum cardinality of 
a set of independent vertices of G, and c(G) be the length of a longest cycle in 
G. A cycle C of G is called a dominating cycle if every edge of G has at least 
one of its vertices on C. We let d(v) denote the degree of vertex v. For k<<,ct(G) 
we let ak(G) = min{~-]ki=ld(vi)l{Vl .....vk} is an independent set of 
k vertices } and NCk(G) be the minimum cardinality of the neighborhood union 
of any k such vertices. If G is a graph on n vertices, then for k > ct(G) we set 
* Corresponding author. 
1 Supported in part by the National Security Agency under Grant MDA 904-89-H-2008. 
2 Supported in part by the National Science Foundation under Grant DMS 9206991. 
0012-365X/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved 
SSD1 0012-365X(94)00364-5  
14 D. Bauer et al./Discrete Mathematics 155 (1996) 13-18 
as(G) = k(n - ~(G)) and NCk(G) = n - ~(G). If G has a noncomplete component we 
let NC2(G) denote the minimum neighborhood union of any pair of vertices at distance 
two apart; otherwise NC2(G) = n - 1. If no ambiguity can arise we sometimes write 
instead of ~(G), etc. 
There are now many results in graph theory relating the toughness of a graph to 
its cycle structure [9,10]. However recently it was discovered [4] that for any fixed 
positive rational number t, it is NP-hard to recognize t-tough graphs. Nevertheless, 
toughness remains an important parameter in the study of hamiltonian cycles in graphs. 
In particular, the truth of Chv~ital's well-known conjecture [16] that 2-tough graphs are 
hamiltonian would have a number of useful applications [1]. On the other hand, the 
fact that it is NP-hard to recognize 1-tough graphs makes it desirable to strengthen 
some theorems by replacing the assumption that a graph G is 1-tough with the weaker 
assumption that G is 2-connected. The idea is to try to draw the same conclusion 
regarding the cycle structure of G under the weaker hypothesis of 2-connectedness by 
specifying an easily described family of exceptional graphs for which the new theorem 
does not hold. 
To illustrate this type of improvement consider the following theorem, established 
independently b  Bondy [12], Bermond [11], and Linial [18]. 
Theorem 1. Let G be a 2-connected 9raph on n vertices. Then c(G)>~ min(n, a2). 
An analogous theorem for 1-tough graphs was later established by Bauer and Schme- 
ichel [8] and independently b  Tian and Zhao [20]. 
Theorem 2. Let G be a 1-touoh 9raph on n>.3 vertices. Then c(G)~ min(n, a2 +2). 
Both of these theorems were generalized in the manner described above by the 
following result, presented three years ago at this conference [5]. We call a graph H 
a az-subgraph of a 2-connected graph G if H is a 2-connected spanning subgraph 
of G with a2(H) = a2(G). Define the graph Sh,p,q to be the graph obtained from 
({u} + pKh) tO ({v} + qKh) by adding the edge uv. Let Th,p,q = K1 + Sh,p,q. 
Theorem 3. Let G be a 2-connected nonhamiltonian 9raph on n vertices with c(G) 
<~ a2 + 1. Then G is a a2-subgraph of one of the followin9: 
(1) K~ + tKl, where t > s >~ 3; 
(2) Ks + (K2 to tK1), where t>>.s>~3; 
(3)  K2 a t- (gl,t to qK2), where q>>.2,t>>.3; 
(4) K3 + qK2, where q>~4; 
(5) K2 + tKh, where t >>. 3, h >>. l ; 
(6) K2 +(tKh UKh_1), where t>~2,h>>.2; 
(7) K2 + (tKh tA Kh+l), where t>~2,h>>. 1; 
(8) Th,p,q, where p>>.2,q>>.2,h>>.l. 
D. Bauer et al. / Discrete Mathematics 155 (1996) 13-18 15 
As another illustration, Skupiefi [19] was able to characterize the nonhamiltonian 2- 
connected graphs with a21> n-  4, thereby improving the following well-known theorem 
of Jung [17]. 
Theorem 4. Let G be a 1-tough graph on n >~ 11 vertices with o-2 >>.n- 4. Then G is 
hamiltonian. 
The purpose of this note is to provide a similar improvement to the following result, 
established in [6]. 
Theorem 5. Let G be a 1-tough graph on n vertices with o-3 >~n. Then every longest 
cycle in G is a dominating cycle. 
An analogous theorem for 2-connected graphs had already been established by Bondy 
[131. 
Theorem 6. Let G be a 2-connected graph on n vertices with o-3 >>- n + 2. Then every 
longest cycle in G is a dominating cycle. 
Our main result, proved in Section 2, generalizes the previous two theorems. 
Consider the following four classes of graphs: 
(1) 1£2 + (Ka U Kb U Kc ) where a,b,c >~ 2; 
(2) K3 + (aK2 U bK3), where a,b>~O and a + b = 4; 
(3) Ks + (sK2 U K3), where s ~> 4; 
(4) Ks+(s+ 1)K2, where s>~4. 
Let ~ i  be the class of all spanning subgraphs of graphs in class (i), 1 ~<i~<4, and 
let ~ = U/4=lJct°i. 
Theorem 7. Let G be a 2-connected graph on n vertices with o-3 >~n and suppose 
G ~ ~.  Then every longest cycle in G is a dominating cycle. 
A number of recent results in hamiltonian graph theory rely on the knowledge that 
every longest cycle in a graph G is a dominating cycle [2,9,10]. Many of these theorems 
employ either Theorem 5 or Theorem 6, i.e., they require that either G is 1-tough with 
o-3 >t n or that G is 2-connected with o-3 >/n + 2. As a consequence of Theorem 7, it is 
no longer necessary to check the NP-hard hypothesis of Theorem 5. 
2. Proof of Theorem 7 
To prove Theorem 7 we make use of a stronger version of Theorem 5. In [6] it was 
noted that the condition that G be 1-tough in Theorem 5 can be replaced by a weaker 
condition. Let ~ol(G) denote the number of components of G having order at least 
16 D. Bauer et al./ Discrete Mathematics 155 (1996) 13-18 
two. A graph G is called co l -  tough if o91(G - S) lSI for every nonempty proper 
subset S of the vertex set V(G). The proof of the following result is implicit in the 
proof of Theorem 5 of [6]. 
Theorem 8. Let G be an Ogl-tOugh graph on n vertices with 63 ~ n. Then every long- 
est cycle in G is a dominating cycle. 
The assumption that G is ~Ol-tOugh in Theorem 8 is a natural one since a necessary 
condition for a graph to have a dominating cycle is that it be col-tough [21]. Theorem 
8 generalizes Theorem 6 and also yields the following simple proof of Theorem 7. 
Proof of Theorem 7. Suppose G satisfies the hypothesis of the theorem and contains 
a longest cycle that is not a dominating cycle. Then by Theorem 8 there exists a 
nonempty set of vertices S such that G - S contains at least IS[ + 1 nontrivial com- 
ponents. Let s = ISI. Since G is 2-connected, s~>2. Let G1,G2 . . . . .  Gs+l+j (j>~0) be 
the nontrivial components of G-  S, and let ni  : IV(Gi)I, 1 <~i<<.s q. 1 q. j .  Let t be 
the number of trivial components of G - S. Assume, without loss of generality, that 
2 ~< nl ~<n2 ~< .-. ~<ns+l+j. I f s : 2, then G E 9(F1 (noting that our characterization is in 
terms of spanning subgraphs). Suppose s>/3. Clearly n >>. n l +n2 q.n3 q," • "q.ns+l+j q-s+ t 
and n<~a3 <<.nl q.n2 q,n3 -3+3s-min(1 , t ) .  Hence 2(sq. j -2)<.. .naq.. . .+ns+l+j<~2s - 
t - min(1,t) - 3, implying that 2j~< 1 - t - min(1,t). We conclude that j = t -- 0 and 
n~+l E {2,3}. Hence i f s  = 3, G E ~2.  Finally, if s~>4, then ns = 2, so G C ~(Y3 if 
n~+l = 3 and G E o~4 if ns+l = 2. [] 
3. Applications 
Let us now return to Theorem 5. The following strengthening of this theorem is also 
in [6]. 
Theorem 9. Let G be a 1-tough graph on n vertices with aa~>n~>3. Then every 
longest cycle in G is a dominating cycle. Moreover, i f  G is nonhamiltonian, G contains 
a longest cycle C such that max{d(v)lv ~ V(G) -  V(C)}>/tr3/3. 
The second part of Theorem 9 is implicit in the proof of [6, Theorem 9]. The details 
of the argument can be found in the appendix of [7]. Using techniques analogous to 
those used to prove Theorem 7 we can also strengthen Theorem 9 so that its hypothesis 
can be checked in polynomial time. 
Theorem 10. Let G be a 2-connected nonhamiltonian graph on n vertices with tr3 >~n 
and suppose G ([ 9f ~. Then G contains a longest cycle C (necessarily dominating) and 
a vertex Vo E V(G) -  V(C) such that d(vo)>>-a3/3 unless G is a spanning subgraph 
o f  a graph in one o f  the following six classes o f  graphs. 
D. Bauer et al./Discrete Mathematics 155 (1996) 13-18 17 
( 1 ) K2 + (Kl U K~ U Kb), where a > b >1 1; 
(2) K3 + (Kl U 2K3 U Ka), where 1 <<. a <~ 3; 
(3) Ks +(K1UK3 U(s -  1)K2), where s>~3," 
(4) Ks + (2K1 UK3 tO (s -  2)K2), where s>>.3," 
(5) Ks + (K1 U sK2), where s>>.3; 
(6) Ks +(2K1U(s -  1)K2), where s>~3. 
As a consequence of Theorems 7 and 10 a number of recent results can be improved 
in the manner previously described. Three of them are given below. 
Theorem 11 (Baner et al. [6]). Let G be a 1-tough graph on n vertices with o-3 >~n>~ 3.
Then c(G)>>, min(n,n + o3/3 - ~). 
Theorem 12 (Baner er al. [3]). Let G be a 1-tough graph on n vertices with o-3 >>.n >~ 3. 
Then c( G ) >~ min(n, 2NC2). 
Theorem 13 (Broersma et al. [14]).. Let G be a 1-tough graph on n vertices with 
a3 >~ n q- r >~ n >~ 3 and n >>. 8k - 6r - 17. Then c( G ) >~ min( n, 2NCk ). 
References 
[1] D. Bauer, H.J. Broersma, J. van den Heuvel and H.J. Veldman, On hamiltonian properties of 2-tough 
graphs, J. Graph Theory 18 (1994) 539-543. 
[2] D. Bauer, H.J. Broersma nd H.J. Veldman, Around three lemmas in hamiltonian graph theory, in: 
R. Bodendiek and R. Henn, ed., Topics in Combinatorics and Graph Theory (Physica - Verlag, 
Heidelberg, 1990) 101-110. 
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A note on dominating cycles in 2-connected graphs

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