We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian

Heuvel, J. van den

English

AbstractWe prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)⩾n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least −k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian

van den Heuvel, J

Elsevier - Publisher Connector 

c~ DISCRETE MATHEMATICS 
ELSEVIER Discrete Mathematics 137 (1995) 389 393 
Note 
Long cycles in graphs containing a 2-factor with 
many odd components 
J. van  den Heuve l  
Faculo' q/" Applied Mathematics, University o[" Twente, P.O. Box 217, 7500 A E Enschede, Netherlands 
Received 13 October 1992; revised 26 February 1993 
Abstract 
We prove a result on the length of a longest cycle in a graph on n vertices that contains 
a 2-factor and satisfies d(u)+ d(v)+d(w)~> n + 2 for every triple u, v, w of independent vertices. 
As a corollary we obtain the following improvement of a conjecture of H/iggkvist (1992): Let 
G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum 
at least n -k  and assume G has a 2-factor with at least k+ 1 odd components. Then G is 
hamiltonian. 
Keywords: (Long, Hamilton) cycle; 2-factor; Degree sum 
1. Results 
We use [4] for terminology and notat ion not defined here and consider finite, 
simple graphs only. 
The following three conjectures, among many others, appear in [6]. 
Conjecture 1. Let G be a 2-connected graph on n vertices where every pair of 
nonadjacent vertices has degree sum at least n -  k and assume furthermore that G has 
a k-factor. Then G is hamiltonian. 
Conjecture 2. Let G be a 2-connected graph on n vertices where every pair of 
nonadjacent vertices has degree sum at least n -k  and assume that G has a 2-factor 
where every component is of order more than k. Then G is hamiltonian. 
Conjecture 3. Let G be a 2-connected graph on n vertices where every pair of 
nonadjacent vertices has degree sum at least n-k  and assume that G has a 2-factor 
with at least 2k odd components. Then G is hamiltonian. 
0012-365X/95/$09.50 ~C) 1995- Elsevier Science B.V. All rights reserved 
SSDI 001 2-365X(93)E01 8-N 
390 J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 
A generalization of Conjecture 1 is proved in [5]. The Petersen graph is a counter- 
example to Conjecture 2, but it is well possible that there are no other counter- 
examples. 
The main goal of this paper is to prove a result which implies that Conjecture 3 is 
true. In fact, our result implies that the bound 2k in Conjecture 3 can be almost halved 
in general. For a graph G and an integer k ~> 1, define ak(G) by 
ak(G)=min {y, d~(v)l S ~_ V(G) is an independent set of size k}. 
v~S 
Now we can state our main result, the proof of which will be given in Section 2. 
Theorem 4. Let G be a 2-connected graph on n vertices that satisfies a3(G)>~n+ 2 and 
assume G contains a 2-factor with at least k odd components. Then a longest cycle in 
G has length at least min{n, X3a3(G)+ ~-±~1"1 
The conclusion in Theorem 4 is best possible. This is shown by the graphs 
Gk3a=Ktv( IK3+(k - l )K2+(t+ l -k )Ka)  (v  denotes the join of two graphs). For 
any k, l, t with k ~> l >~ 1 and t ~> 21 + k + 2, Gk,~,~ is a 2-connected graph on n = 2t + 21 + k 
vertices that contains a 2-factor with k odd components, but no 2-factor with more 
than k odd components, and satisfies a3(Gk, t,t)=3t>~n+2. The length of a longest 
cycle in Gk,/,, is 31+2(k - l )+( t -k )+t=2t+l+k=½a3(Gk .~. , )+½n+½k.  
Also the bound a3(G)~>n+2 in Theorem 4 cannot be relaxed as is shown by the 
graphs  /-/t = Kz v (K3t + K31 + K31_ 2)- For any I ~> 1, H~ is a 2-connected graph on 
n=91 vertices that has a 2-factor with 31 odd components and satisfies 
a3(H, )=91+l=n+l .  Furthermore, ~a3(Hz)+½n+½(31)=9l+½, whereas a longest 
cycle in H~ has length 61 + 2 only. 
From Theorem 4 we can derive the following corollaries concerning Hamilton 
cycles in graphs. Corollary 7 shows that (a sharper version of) Conjecture 3 is 
true. 
Corollary 5. Let G be a 2-connected graph on n vertices that satisfies ~r3(G)~> 
max {23(n - k ) -  1, n + 2} and contains a 2-factor with at least k - 1 odd components. Then 
G is hamiltonian. 
Proof. Let G be a graph that satisfies the conditions in the corollary. Then we have 
~a3(G)÷½n+½(k-  >-' ' ' 1 )~.2(n -k ) -~+2n+½(k-  1)=n-~> n-  1, 
so, by Theorem 4 we can conclude that G contains a Hamilton cycle. [] 
Corollary 6. Let G be a 2-connected graph on n vertices that satisfies az(G)>~ 
max {n-  k, Zn + 1 } and contains a 2-factor with at least k -  1 odd components. Then G is 
hamiltonian. 
J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 391 
Proof. For any graph G and any three independent vertices u, v, w in V(G) we have 
d(u) + d (v) + d(w) = ½ [(d(u) + d (v) ) + ( d u) + d(w) + ( d(v) + d(w))] 
~> ½(3o2(G)). 
So, for any graph G, o'3(G)~>3O'2(G) and Corollary 6 follows immediately from 
Corollary 5. [] 
Corollary 7. Let G be a 2-connected graph on n vertices that satisfies a2(G)>~ n-k  and 
contains a 2-factor with at least k + 1 odd components. The G is hamiltonian. 
Proof. Let G be graph that satisfies the conditions in the corollary. Since G contains 
a 2-factor with at least k + 1 odd components, we have n ~> 3(k + 1), or k ~<~n- 1. This 
means a2(G)~n-k>~n+l ,  and the result follows from Corollary 6. [Z 
Notice that for k~<~(n-5) we have V~- (n -k ) - l ] />n+2-  So the bound 
a3(G)>~max{3(n-k) - 1,n+2} in Corollary 5 can be replaced by a3(G)>~3(n-k) - 1 
if we add the extra condition k~<½(n-5). Analogously, the bound a2(G)~ 
max{n-k ,2n+l}  in Corollary 6 can be replaced by a2(G)>~n-k if we add the 
condition k~<](n-3). This is not a strong limitation, because we already have the 
condition that G contains a 2-factor with at least k -  1 odd components, which means 
n~>3(k-  1), or k~<~(n+3). 
The graphs Gk- 1, 1. =Kt  v (K3 +(k -2)K2  +(t -k+2)K1)  show that the bounds in 
Corollaries 5 and 6 are sharp. For any k,t with k~>2 and t>~k+3, Gk-Ll,t is 
a 2-connected graph on n = 2t + k + 1 vertices that contains a 2-factor with k -  1 odd 
components, satisfies a3(Gk_ l , l , t )=3t=3(n-k ) - l -½ and a2(Gk_l,l,t)=2t= 
n- -k -1 ,  but contains no Hamilton cycle. 
2. Proof of Theorem 4 
Let C be a cycle of a graph G. If V(G)-  V(C) is a independent set, then C is called 
a dominating cycle of G. By ~ we denote the cycle C with a given orientation. If 
u e V(C), then u + denotes the successor of u on ~. If A _~ V(C), then A + = { v + [ v e A }. 
The following two lemmas are essential in our proof of Theorem 4. The first part 
of Lemma 8 is a result from [3]; the second part is implicit in the proof of [2, 
Theorem 10]. Lemma 9 is [2, Lemma 8]. Both lemmas also appear in [1]. 
Lemma 8 (Bauer et al. [1,2], Bondy [33). Let G be a 2-connected graph on n vertices 
that satisfies aa(G)~>n+2. Then every longest cycle of G is a dominating cycle. 
Moreover, if G is nonhamihonian, then G contains a longest cycle C such that 
max{d~(v) [ v~ V(G)-  V(C)} ~>~a3(G). 
392 J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 
Lemma 9 (Bauer et al. [-1,2]). Let G be a 9raph on n vertices with 6(G)>~2 and 
tr3(G)~>n. Assume that G contains a lonyest cycle ~ which is a dominatin9 cycle. If 
vE V(G)-  V(C), then (V(G)-  V(C)) w (N(v)) + is an independent set of vertices. 
For the remainder of this section we assume that G is a nonhamiltonian, 2- 
connected graph on n vertices that satisfies tr3(G)>~n+2 and contains a 2-factor 
F with at least k odd components. By Lemma 8, we can choose a longest cycle C in 
G and a vertex at  V(G)-  V(C) such that N(a) c_ V(C) and d~(a)>~½a3(G). Let c be the 
length of C and choose an orientation C of C. Define A = (V(G) -- V(C)) w (N (a)) + and 
B= V(G)--A. By Lemmas 8 and 9, A is an independent set of vertices of size 
]Al>~n--c + ½cr3(G). 
Since A is an independent set, all edges in G either have one end vertex in A 
and the other end vertex in B, or both end vertices in B. Of course the same holds for 
the edges of the 2-factor F. Let er(B) be the number of edges of F with both end 
vertices in B. Every odd component of F contains at least one edge with both end 
vertices in B, hence er(B)>>.k. Counting the edges of F between A and B in two ways 
we see 
2]A] =21B]--2ev(B). 
Since ]BI = n -  I A ], this is equivalent to 
41A ] = 2n-  2ev(B). 
Using that ]AI >>.n-c+~tr3(G) and ev(B)>~k we obtain 
4a3(G) + 4n-4c  ~< 2n-  2k, 
which gives 
This completes the proof of Theorem 4. 
Acknowledgment 
I thank H.J. Veldman for his help and suggestions during the preparation of this 
paper. 
References 
[1] D. Bauer, H.J. Broersma nd H.J. Veldman, Around three lemmas in hamiltonian graph theory, in: 
R. Bodendiek and R. Henn, eds, Topics in Combinatorics and Graph Theory, Essays in Honour of 
Gerhard Ringel (Physica-Verlag, Heidelberg, 1990) 101-110. 
[2] D. Bauer, A. Morgana, E.F. Schmeichel and H.J. Veldman, Long cycles in graphs with large degree 
sums, Discrete Math. 79 (1989/90) 59-70. 
J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 393 
[3] J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, Univ. of 
Waterloo, Waterloo, Ontario (1980). 
[4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, 
New York, 1976). 
[5] R.J. Faudree and J. van den Heuvel, Degree sums, k-factors and Hamilton cycles in graphs, preprint 
(1992). 
[6] R. H~iggkvist, Twenty odd statements in Twente, in: H.J. Broersma, J. van den Heuvel and H,J. 
Veldman, eds, Updated Contributions to the Twente Workshop on Hamiltonian Graph Theory, 6-10 
April 1992. Univ. of Twente, Enschede, Netherlands (1992) 67-76. 


Long cycles in graphs containing a 2-factor with many odd components 

van den Heuvel, J.

NARCIS 

Long cycles in graphs containing a 2-factor with many odd components

University of Twente Research Information

c ~  DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 137 (1995) 389 393 Note Long cycles in graphs containing a 2-factor with many odd components J. v a n  d e n  H e u v e l  Faculo' q/" Applied Mathematics, University o[" Twente, P.O. Box 217, 7500 A E Enschede, Netherlands Received 13 October 1992; revised 26 February 1993 Abstract We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+ d(v)+d(w)~> n + 2 for every triple u, v, w of independent vertices. As a corollary we obtain the following improvement of a conjecture of H/iggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n - k  and assume G has a 2-factor with at least k+ 1 odd components. Then G is hamiltonian. Keywords: (Long, Hamilton) cycle; 2-factor; Degree sum 1. Results We use [4] for terminology and no ta t ion  not  defined here and consider finite, simple graphs only. The following three conjectures, among  many  others, appear in [6]. Conjecture 1. Let G be a 2-connected graph on n vertices where every pair of nonad jacen t  vertices has degree sum at least n -  k and assume furthermore that G has a k-factor. Then  G is hamil tonian.  Conjecture 2. Let G be a 2-connected graph on n vertices where every pair of nonad jacen t  vertices has degree sum at least n - k  and assume that G has a 2-factor where every componen t  is of order more than k. Then G is hamil tonian.  Conjecture 3. Let G be a 2-connected graph on n vertices where every pair of nonad jacen t  vertices has degree sum at least n - k  and  assume that G has a 2-factor with at least 2k odd components .  Then  G is hamil tonian.  0012-365X/95/$09.50 ~C) 1995- Elsevier Science B.V. All rights reserved SSDI 001 2 -365X(93)E01  18-N 390 J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 A generalization of Conjecture 1 is proved in [5]. The Petersen graph is a counter- example to Conjecture 2, but it is well possible that there are no other counter- examples. The main goal of this paper is to prove a result which implies that Conjecture 3 is true. In fact, our result implies that the bound 2k in Conjecture 3 can be almost halved in general. For a graph G and an integer k ~> 1, define ak(G) by ak(G)=min { y, d~(v)l S ~_ V(G) is an independent set of size k}. v ~ S  Now we can state our main result, the proof of which will be given in Section 2. Theorem 4. Let G be a 2-connected graph on n vertices that satisfies a3(G)>~n+ 2 and assume G contains a 2-factor with at least k odd components. Then a longest cycle in G has length at least min{n,  X3a3(G)+ ~-±~1"1 The conclusion in Theorem 4 is best possible. This is shown by the graphs G k 3 a = K t v ( I K 3 + ( k - l ) K 2 + ( t + l - k ) K a )  ( v  denotes the join of two graphs). For any k, l, t with k ~> l >~ 1 and t ~> 21 + k + 2, Gk,~,~ is a 2-connected graph on n = 2t + 21 + k vertices that contains a 2-factor with k odd components, but no 2-factor with more than k odd components, and satisfies a3(Gk, t , t)=3t>~n+2. The length of a longest cycle in Gk,/,, is 3 1 + 2 ( k - l ) + ( t - k ) + t = 2 t + l + k = ½ a 3 ( G k . ~ . , ) + ½ n + ½ k .  Also the bound a3(G)~>n+2 in Theorem 4 cannot be relaxed as is shown by the g r a p h s  /-/t = Kz v (K3t + K31 + K31_ 2)- For any I ~> 1, H~ is a 2-connected graph on n=91 vertices that has a 2-factor with 31 odd components and satisfies a 3 ( H , ) = 9 1 + l = n + l .  Furthermore, ~a3(Hz)+½n+½(31)=9l+½, whereas a longest cycle in H~ has length 61 + 2 only. From Theorem 4 we can derive the following corollaries concerning Hamilton cycles in graphs. Corollary 7 shows that (a sharper version of) Conjecture 3 is true. Corollary 5. Let G be a 2-connected graph on n vertices that satisfies ~r3(G)~> max {23(n - k ) -  1, n + 2} and contains a 2-factor with at least k - 1 odd components. Then G is hamiltonian. Proof. Let G be a graph that satisfies the conditions in the corollary. Then we have ~ a 3 ( G ) ÷ ½ n + ½ ( k -  >-' ' ' 1 ) ~ . 2 ( n - k ) - ~ + 2 n + ½ ( k -  1 ) = n - ~ >  n -  1, so, by Theorem 4 we can conclude that G contains a Hamilton cycle. [] Corollary 6. Let G be a 2-connected graph on n vertices that satisfies az(G)>~ max { n -  k, Zn + 1 } and contains a 2-factor with at least k -  1 odd components. Then G is hamiltonian. J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 391 Proof. For any graph G and any three independent vertices u, v, w in V(G) we have d(u) + d (v) + d(w) = ½ [(d(u) + d (v) ) + ( d(u) + d(w) ) + ( d(v) + d(w))] ~> ½(3o2(G)). So, for any graph G, o'3(G)~>3O'2(G) and Corollary 6 follows immediately from Corollary 5. [] Corollary 7. Let G be a 2-connected graph on n vertices that satisfies a2(G)>~ n - k  and contains a 2-factor with at least k + 1 odd components. The G is hamiltonian. Proof. Let G be graph that satisfies the conditions in the corollary. Since G contains a 2-factor with at least k + 1 odd components,  we have n ~> 3(k + 1), or k ~<~n- 1. This means a 2 ( G ) ~ n - k > ~ n + l ,  and the result follows from Corollary 6. [Z Notice that for k~<~(n-5)  we have V ~ - ( n - k ) - l ] / > n + 2 -  So the bound a3(G)>~max{3(n-k) - 1,n+2} in Corollary 5 can be replaced by a3(G)>~3(n-k) - 1 if we add the extra condition k~<½(n-5). Analogously, the bound a2(G)~ m a x { n - k , 2 n + l }  in Corollary 6 can be replaced by a2(G)>~n-k if we add the condition k~<](n-3) .  This is not a strong limitation, because we already have the condition that G contains a 2-factor with at least k -  1 odd components,  which means n~>3(k-  1), or k~<~(n+3). The graphs Gk- 1, 1. = K t  v (K3 + ( k - 2 ) K 2  + ( t - k + 2 ) K 1 )  show that the bounds in Corollaries 5 and 6 are sharp. For any k,t with k~>2 and t>~k+3, Gk-Ll , t  is a 2-connected graph on n = 2t + k + 1 vertices that contains a 2-factor with k -  1 odd components, satisfies a 3 ( G k _ l , l , t ) = 3 t = 3 ( n - k ) - l - ½  and a2(Gk_l, l , t )=2t= n - - k - 1 ,  but contains no Hamil ton cycle. 2. Proof of Theorem 4 Let C be a cycle of a graph G. If V(G) -  V(C) is a independent set, then C is called a dominating cycle of G. By ~ we denote the cycle C with a given orientation. If u e V(C), then u + denotes the successor of u on ~.  If A _~ V(C), then A + = { v + [ v e A }. The following two lemmas are essential in our proof  of Theorem 4. The first part  of Lemma 8 is a result from [3]; the second part is implicit in the proof  of [2, Theorem 10]. Lemma 9 is [2, Lemma 8]. Both lemmas also appear in [1]. Lemma 8 (Bauer et al. [1,2], Bondy [33). Let G be a 2-connected graph on n vertices that satisfies aa(G)~>n+2. Then every longest cycle of G is a dominating cycle. Moreover, if G is nonhamihonian, then G contains a longest cycle C such that max{d~(v) [ v~ V(G) -  V(C)} ~>~a3(G). 392 J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 Lemma 9 (Bauer et al. [-1,2]). Let G be a 9raph on n vertices with 6(G)>~2 and tr3(G)~>n. Assume that G contains a lonyest cycle ~ which is a dominatin9 cycle. I f  vE V(G) -  V(C), then ( V ( G ) -  V(C)) w (N(v)) + is an independent set of vertices. For the remainder of this section we assume that G is a nonhamiltonian, 2- connected graph on n vertices that satisfies tr3(G)>~n+2 and contains a 2-factor F with at least k odd components. By Lemma 8, we can choose a longest cycle C in G and a vertex a t  V (G) -  V(C) such that N(a) c_ V(C) and d~(a)>~½a3(G). Let c be the length of C and choose an orientation C of C. Define A = (V(G) -- V(C)) w (N (a)) + and B= V(G)--A. By Lemmas 8 and 9, A is an independent set of vertices of size ]Al>~n--c + ½cr3(G). Since A is an independent set, all edges in G either have one end vertex in A and the other end vertex in B, or both end vertices in B. Of course the same holds for the edges of the 2-factor F. Let er(B) be the number of edges of F with both end vertices in B. Every odd component  of F contains at least one edge with both end vertices in B, hence er(B)>>.k. Counting the edges of F between A and B in two ways we see 2]A] =21B]--2ev(B). Since ]BI = n -  I A ], this is equivalent to 41A ] = 2 n -  2ev(B). Using that ]AI >>.n-c+~tr3(G) and ev(B)>~k we obtain 4a3(G) + 4 n - 4 c  ~< 2 n -  2k, which gives This completes the proof of Theorem 4. Acknowledgment I thank H.J. Veldman for his help and suggestions during the preparation of this paper. References [1] D. Bauer, H.J. Broersma and H.J. Veldman, Around three lemmas in hamiltonian graph theory, in: R. Bodendiek and R. Henn, eds, Topics in Combinatorics and Graph Theory, Essays in Honour of Gerhard Ringel (Physica-Verlag, Heidelberg, 1990) 101-110. [2] D. Bauer, A. Morgana, E.F. Schmeichel and H.J. Veldman, Long cycles in graphs with large degree sums, Discrete Math. 79 (1989/90) 59-70. J. van den Heuvel/ Discrete Mathematics 137 (1995) 389-393 393 [3] J.A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, Univ. of Waterloo, Waterloo, Ontario (1980). [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976). [5] R.J. Faudree and J. van den Heuvel, Degree sums, k-factors and Hamilton cycles in graphs, preprint (1992). [6] R. H~iggkvist, Twenty odd statements in Twente, in: H.J. Broersma, J. van den Heuvel and H,J. Veldman, eds, Updated Contributions to the Twente Workshop on Hamiltonian Graph Theory, 6-10 April 1992. Univ. of Twente, Enschede, Netherlands (1992) 67-76. 

Around three lemmas in hamiltonian graph theory, in:

Degree sums, k-factors and Hamilton cycles in graphs, preprint

Graph Theory with Applications (Macmillan, London and Elsevier,

Long cycles in graphs with large degree sums,

Longest paths and cycles in graphs of high degree,

Twenty odd statements in Twente, in:

file:///data/core-remote/dit/data/elsevier/pdf/243/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDkzZTAxMThu.pdf

Long cycles in graphs containing a 2-factor with many odd components

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