On path-quasar Ramsey numbers

Binlong Li, Bo Ning

Abstract


Let \(G_1\) and \(G_2\) be two given graphs. The Ramsey number \(R(G_1,G_2)\) is the least integer \(r\) such that for every graph \(G\) on \(r\) vertices, either \(G\) contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). Parsons gave a recursive formula to determine the values of \(R(P_n,K_{1,m})\), where \(P_n\) is a path on \(n\) vertices and \(K_{1,m}\) is a star on \(m+1\) vertices. In this note, we study the Ramsey numbers \(R(P_n,K_1\vee F_m)\), where \(F_m\) is a linear forest on \(m\) vertices. We determine the exact values of \(R(P_n,K_1\vee F_m)\) for the cases \(m\leq n\) and \(m\geq 2n\), and for the case that \(F_m\) has no odd component. Moreover, we give a lower bound and an upper bound for the case \(n+1\leq m\leq 2n-1\) and \(F_m\) has at least one odd component.

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References


Bondy, J. A., Murty, U. S. R., Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.

Chen, Y., Zhang, Y., Zhang, K., The Ramsey numbers of paths versus wheels, Discrete Math. 290 (1) (2005), 85–87.

Dirac, G. A., Some theorems on abstract graphs, Proc. London. Math. Soc. 2 (1952), 69–81.

Faudree, R. J., Lawrence, S. L., Parsons, T. D., Schelp, R. H., Path-cycle Ramsey numbers, Discrete Math. 10 (2) (1974), 269–277.

Graham, R. L., Rothschild, B. L., Spencer, J. H., Ramsey Theory, Second Edition, John Wiley & Sons Inc., New York, 1990.

Li, B., Ning, B., The Ramsey numbers of paths versus wheels: a complete solution, Electron. J. Combin. 21 (4) (2014), #P4.41.

Parsons, T. D., Path-star Ramsey numbers, J. Combin. Theory, Ser. B 17 (1) (1974), 51–58.

Rousseau, C. C., Sheehan, J., A class of Ramsey problems involving trees, J. London Math. Soc. 2 (3) (1978), 392–396.

Salman, A. N. M., Broersma, H. J., Path-fan Ramsey numbers, Discrete Applied Math. 154 (9) (2006), 1429–1436.

Salman, A. N. M., Broersma, H. J., Path-kipas Ramsey numbers, Discrete Applied Math. 155 (14) (2007), 1878–1884.

Zhang, Y., On Ramsey numbers of short paths versus large wheels, Ars Combin. 89 (2008), 11–20.




DOI: http://dx.doi.org/10.17951/a.2014.68.2.11
Date of publication: 2015-05-23 16:29:44
Date of submission: 2015-05-09 13:05:20


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