A Ramsey problem related to butterfly graph vs. proper connected subgraphs of K4

TL;DR

This paper determines the exact multipartite Ramsey numbers for the Butterfly graph versus connected proper subgraphs of K4, advancing understanding of edge-coloring problems involving specific small graphs.

Contribution

It provides exact values of the multipartite Ramsey number mj(B,G) for all connected proper subgraphs G of K4, a new result in graph Ramsey theory.

Findings

Exact values of mj(B,G) for all connected proper subgraphs G of K4.

Characterization of edge-colorings avoiding monochromatic Butterfly graphs.

Extension of multipartite Ramsey theory to specific small graphs.

Abstract

A graph on 5 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 exact the values for mj(B,G) when and G represents any connected proper subgraph of K4 with at least one edge.