A Ramsey Problem Related to Butterfly Graph vs. Small Paths and C3

Abstract

A graph on five vertices consisting of 2 copies of the cycle graph C3 sharing a common vertex is called the Butterfly graph (B). The smallest natural number s such that any two-colouring (say red and blue) of the edges of Kj×s has a copy of a red B or a blue G is called the multipartite Ramsey number of Butterfly graph versus G. This number is denoted by mj(B,G). In this paper, we find the exact values for mj(B,G) when j >2 and G represents any small path or else three cycles.