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The minimum degree, connectivity and independence number of G are denoted by δ(G), κ(G) and α(G), respectively

# The Ramsey numbers for cycles versus wheels of odd order

Applied Mathematics Letters, no. 12 (2009): 1875-1876

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Abstract

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Cn denote a cycle of order n and Wm a wheel of order m+1. It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that R(Cn,Wm)=2n−1 for even m≥4, n≥m an...More

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Introduction
• Let G = (V (G), E(G)) be a graph.
• The authors use Cn and mKn to denote a cycle of order n and the union of m vertex disjoint complete graphs Kn, respectively.
• A Wheel Wm = K1 + Cm is a graph of m + 1 vertices.
• A graph G is Hamilton-connected if it contains a Hamiltonian path between any two distinct vertices.
Highlights
• The minimum degree, connectivity and independence number of G are denoted by δ(G), κ(G) and α(G), respectively
• We use Cn and mKn to denote a cycle of order n and the union of m vertex disjoint complete graphs Kn, respectively
• A Wheel Wm = K1 + Cm is a graph of m + 1 vertices
• The lengths of the longest and shortest cycles of G are denoted by c(G) and g(G), respectively
• Let G be a graph of order 2n − 1 without Cn
Results
• If G has an edge-induced subgraph Gi, the authors say G contains Gi. For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest integer n such that for any graph G of order n, either G
• Some Ramsey numbers concerning cycles versus wheels have been obtained; see [1] for details.
• Considered the Ramsey number R(Cn, Wm) in the case when n ≥ m and established the following.
• For Conjecture 1, Surahmat et al [3] obtained the following partial result.
• Theorem 2 (Surahmat et al [3]).
• Y. Chen et al / Applied Mathematics Letters 22 (2009) 1875–1876
• By using the technique of dominating cycles, Zhang et al [4] established the following result.
• The authors consider the Ramsey number R(Cn, Wm) in the case when m is even.
• In order to prove Theorem 4, the authors need the following lemmas.
• Every non-bipartite graph G of order n with δ(G) ≥ (n + 2)/3 is weakly pancyclic with g(G) = 3 or 4.
• Let G be a 2-connected graph of order n ≥ 3 with δ(G) = δ.
• Lemma 5 (Surahmat et al [3]).
• Let G be a graph of order 2n − 1 without Cn. Suppose m ≥ 4 is even and n ≥ 3m/2.
Conclusion
• Let G be a graph of order 2n − 1.
• By Lemma 2, G is weakly pancyclic of girth 3 or 4.
• If κ(G) ≥ 2, c(G) ≥ min{2n − m, 2n − 1} > n by Lemma 3, which implies that G contains a Cn, a contradiction.
• By Lemma 1, G contains a Cn, a contradiction.
Summary
• Let G = (V (G), E(G)) be a graph.
• The authors use Cn and mKn to denote a cycle of order n and the union of m vertex disjoint complete graphs Kn, respectively.
• A Wheel Wm = K1 + Cm is a graph of m + 1 vertices.
• A graph G is Hamilton-connected if it contains a Hamiltonian path between any two distinct vertices.
• If G has an edge-induced subgraph Gi, the authors say G contains Gi. For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest integer n such that for any graph G of order n, either G
• Some Ramsey numbers concerning cycles versus wheels have been obtained; see [1] for details.
• Considered the Ramsey number R(Cn, Wm) in the case when n ≥ m and established the following.
• For Conjecture 1, Surahmat et al [3] obtained the following partial result.
• Theorem 2 (Surahmat et al [3]).
• Y. Chen et al / Applied Mathematics Letters 22 (2009) 1875–1876
• By using the technique of dominating cycles, Zhang et al [4] established the following result.
• The authors consider the Ramsey number R(Cn, Wm) in the case when m is even.
• In order to prove Theorem 4, the authors need the following lemmas.
• Every non-bipartite graph G of order n with δ(G) ≥ (n + 2)/3 is weakly pancyclic with g(G) = 3 or 4.
• Let G be a 2-connected graph of order n ≥ 3 with δ(G) = δ.
• Lemma 5 (Surahmat et al [3]).
• Let G be a graph of order 2n − 1 without Cn. Suppose m ≥ 4 is even and n ≥ 3m/2.
• Let G be a graph of order 2n − 1.
• By Lemma 2, G is weakly pancyclic of girth 3 or 4.
• If κ(G) ≥ 2, c(G) ≥ min{2n − m, 2n − 1} > n by Lemma 3, which implies that G contains a Cn, a contradiction.
• By Lemma 1, G contains a Cn, a contradiction.
Funding
• This research was supported by NSFC under grant number 10671090, 10871166 and in part by the Research Grants Council of Hong Kong under grant number PolyU5136/08E
• Miao was also supported in part by NSF of Jiangsu Province under grant number 07KJD110207 and NSF of University in Jiangsu Province under grant number BK2007030
Reference
• S.P. Radziszowski, Small Ramsey numbers, The Electronic Journal of Combinatorics (2006) DS1.11.
• Surahmat, E.T. Baskoro, I. Tomescu, The Ramsey numbers of large cycles versus wheels, Discrete Mathematics 306 (2006) 3334–3337.
• Surahmat, E.T. Baskoro, I. Tomescu, The Ramsey numbers of large cycles versus odd wheels, Graphs and Combinatorics 24 (2008) 53–58.
• L.M. Zhang, Y.J. Chen, T.C. Edwin Cheng, The Ramsey numbers for cycles versus wheels of even order, European Journal of Combinatorics (in press).
• J.A. Bondy, Pancyclic graphs, Journal of Combinatorial Theory, Series B 11 (1971) 80–84.
• S. Brandt, R.J. Faudree, W. Goddard, Weakly pancyclic graphs, Journal of Graph Theory 27 (1998) 141–176.
• G.A. Dirac, Some theorems on abstract graphs, Proceedings of the London Mathematical Society 2 (3) (1952) 69–81.
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