On Multicolor Ramsey Number of Paths Versus Cycles

Let G1, G2, . . . , Gt be graphs. The multicolor Ramsey number R(G1, G2, . . . , Gt) is the smallest positive integer n such that if the edges of a complete graph Kn are partitioned into t disjoint color classes giving t graphs H1,H2, . . . ,Ht, then at least one Hi has a subgraph isomorphic to Gi. In this paper, we provide the exact value of R(Pn1 , Pn2 , . . . , Pnt , Ck) for certain values of ni and k. In addition, the exact values of R(P5, C4, Pk), R(P4, C4, Pk), R(P5, P5, Pk) and R(P5, P6, Pk) are given. Finally, we give a lower bound for R(P2n1 , P2n2 , . . . , P2nt) and we conjecture that this lower bound is the exact value of this number. Moreover, some evidence is given for this conjecture.