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# 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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