Rainbow subgraphs of star-coloured graphs

TL;DR

This paper investigates the maximum number of colours in star-coloured complete graphs that avoid rainbow copies of a specific graph, extending previous work on generalized Ramsey numbers.

Contribution

It extends existing results on rainbow subgraphs in star-coloured graphs, providing new bounds for complete graphs avoiding certain rainbow subgraphs.

Findings

Determined maximum colours in star-coloured complete graphs without rainbow H

Extended results on generalized Ramsey numbers for star-coloured graphs

Provided bounds for rainbow subgraphs in large complete graphs

Abstract

An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the first structure, and a star-colouring if it forbids the second structure. In this paper, we study rainbow subgraphs in star-coloured graphs and determine the maximum number of colours in a star-colouring of a large complete graph which does not contain a rainbow copy of a given graph $H$. This problem is a special case of one studied by Axenovich and Iverson on generalised Ramsey numbers and we extend their results in this case.