Star-path and star-stripe bipartite Ramsey numbers in multicoloring

Document Type : Research Paper


Department of Mathematical Sciences, Shahrekord University, P. O. Box 115, Shahrekord, Iran


‎‎For given bipartite graphs $G_1‎, ‎G_2,\ldots‎, ‎G_t,$ the bipartite Ramsey number $bR(G_1‎, ‎G_2,\ldots‎, ‎G_t)$ is the‎ ‎smallest integer $n$ such that if the edges of the complete bipartite graph $K_{n,n}$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1‎, ‎H_2,\ldots‎, ‎H_t$‎, ‎then at least one $H_i$ has a subgraph isomorphic to $G_i$‎. ‎In this paper‎, ‎we study the multicolor bipartite Ramsey number $bR(G_1‎, ‎G_2,\ldots‎, ‎G_t)$‎, ‎in the case that $G_1‎, ‎G_2,\ldots‎, ‎G_t$ being either stars and stripes or stars and a path‎.


Main Subjects

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

[2] P. Erdos and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc, 62 (1956) 427-489.

[3] R. J. Faudree and R. H. Schelp, Path-path Ramsey-type numbers for the complete bipartite graph, J. Combinatorial Theory Ser. B, 19  (1975) 161-173.

[4] A. Gyarfas and J. Lehel, A Ramsey-type problem in directed and bipartite graphs, Period. Math. Hungar., 3 (1973) 299-304.

[5] J. H. Hattingh and M. A. Henning, Star-path bipartite Ramsey numbers, Discrete Math., 185  (1998) 255-258.

[6] J. H. Hattingh and M. A. Henning, Bipartite Ramsey theory, Util. Math., 53  (1998) 217-230.

[7] R. W. Irving, A bipartite Ramsey problem and the Zarankiewicz numbers, Glasgow Math. J., 19 (1978) 13-26.

[8] S. P. Radziszowski, Small Ramsey numbers, Electronic J. Combin., (1994) pp. 30, Dynamic Survey 1.