On size multipartite Ramsey numbers for stars versus paths and cycles

Abstract

Let $K_{l\times t}$ be a complete, balanced, multipartite graph consisting of $l$ partite sets and $t$ vertices in each partite set. For given two graphs $G_1$ and $G_2$, and integer $j\geq 2$, the size multipartite Ramsey number $m_j(G_1,G_2)$ is the smallest integer $t$ such that every factorization of the graph $K_{j\times t}:=F_1\oplus F_2$ satisfies the following condition: either $F_1$ contains $G_1$ or $F_2$ contains $G_2$. In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths $P_n$ versus stars, for $n=2,3$ only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths $P_n$ versus stars, for $n=3,4,5,6$. In this paper, we investigate the size tripartite Ramsey numbers of paths $P_n$ versus stars, with all $n\geq 2$. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers $m_2(K_{1,m},C_n)$ of stars versus cycles, for $n\geq 3,m\geq 2$.