Rainbow subgraphs of star-coloured graphs

TL;DR

This work initiates and develops the study of rainbow subgraphs in star-coloured complete graphs, focusing on the regime where the target graph $H$ has vertex arboricity ${\mathrm{va}}(H)\le 2$ and introducing the star-anti-Ramsey number ${\mathrm{ar}}^{\star}(n,H)$. It establishes sharp results for cycles ${C_k}$ and several small graphs (notably ${K_3}$, ${K_4}$, ${K_4^-}$, ${K_5^-}$), and provides general bounds for joins of trees via connections to the Zarankiewicz problem and dependent random choice. The paper also develops a toolkit of special and modified colourings (lexical, orientable, rainbow blow-ups) that yield both lower and upper bounds and characterise extremal colourings in several cases. Finally, it outlines open problems on star-realisable densities and linear star-anti-Ramsey numbers, highlighting rich directions for future extremal-graph-theory research.

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.

Figures (1)

• Figure 1: Extremal star-colourings for $K_4^-$. On the left, $n=9$; on the right, $n=10$, $a=2$ and the edges of $S$ are yellow.

Theorems & Definitions (77)

• Theorem 1.1: The canonical Ramsey theorem ErdosRado
• Theorem 1.2: ErdosStoneKovariSosTuran
• Theorem 1.3: AxenovichIverson
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.10
• Theorem 2.1