Rainbow cycles through specified vertices

TL;DR

This paper studies the minimum number of colours needed to edge-colour graphs so that every set of k vertices lies in a rainbow cycle, providing results for general graphs and specific graph classes.

Contribution

It introduces the concept of the k-rainbow cycle index and characterizes this parameter for various classes of graphs, including a classification for when it equals the number of edges.

Findings

Classified graphs with $crx_k(G)=e(G)$ for $k=1,2$

Determined $crx_k(G)$ for wheels, complete, bipartite, multipartite graphs, and discrete cubes

Provided general results about $crx_k(G)$ for arbitrary graphs

Abstract

An edge-coloured cycle is rainbow if the edges have distinct colours. Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required to colour the edges of $G$ such that, for every set $S$ of $k$ vertices in $G$, there exists a rainbow cycle in $G$ containing $S$. In this paper, we will first prove some results about the parameter $crx_k(G)$ for general graphs $G$. One of the results is a classification of all graphs $G$ such that $crx_k(G)=e(G)$, for $k=1,2$. We will also determine $crx_k(G)$ for some specific graphs $G$, including wheels, complete graphs, complete bipartite and multipartite graphs, and discrete cubes.