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

Abstract

‎‎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.