On Multicolor Ramsey Number of Paths Versus Cycles

Abstract

Let $G_1, G_2, G_3, \ldots , G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of a complete graph $K_n$ 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 provide the exact value of $R(P_{n_1}, P_{n_2},\ldots, P_{n_t},C_k)$ for certain values of $n_i$ and $k$. In addition, the exact values of $R(P_5,C_4,P_k)$, $R(P_4,C_4,P_k)$, $R(P_5,P_5,P_k)$ and $R(P_5,P_6,P_k)$ are given. Finally, we give a lower bound for $R(P_{2n_1}, P_{2n_2},\ldots, P_{2n_t})$ and we conjecture that this lower bound is the exact value of this number. Moreover, some evidence is given for this conjecture.