List rainbow connection number of graphs

TL;DR

This work introduces the list rainbow connection numbers $rc^{\ell}(G)$ and list strong rainbow connection numbers $src^{\ell}(G)$, establishing foundational inequalities that relate them to the ordinary parameters $rc(G)$ and $src(G)$ and exploring their behavior across graph classes. It provides general bounds, characterizations for trees and complete graphs, and universal-vertex scenarios, and it derives exact values or tight bounds for cycles, wheels, complete bipartite and multipartite graphs, and the Petersen graph in the list-coloring setting. The authors also characterize which pairs of values can occur for $(src(G),src^{\ell}(G))$ and $(rc^{\ell}(G),src^{\ell}(G))$, with constructive proofs showing that for $a,b$ with $2\le a\le b$ (or $a=b=1$) such graphs exist; they pose the central open problem of whether $rc(G)=rc^{\ell}(G)$ for all connected graphs. Overall, the paper advances understanding of list-edge-colorings in rainbow connectivity and lays groundwork for further study of list analogues of classic rainbow parameters.

Abstract

An edge-coloured path is rainbow if all of its edges have distinct colours. Let $G$ be a connected graph. The rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow path. The strong rainbow connection number of $G$, denoted by $src(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two notions of connectivity of graphs were introduced by Chartrand, Johns, McKeon and Zhang in 2008. In this paper, we introduce the list rainbow connection number $rc^\ell(G)$, and the list strong rainbow connection number $src^\ell(G)$. These two parameters are the versions of $rc(G)$ and $src(G)$ that involve list edge-colourings. Among our results, we will determine the list rainbow connection number and list strong rainbow connection number of some specific graphs. We will also characterise all pairs of positive integers $a$ and $b$ such that, there exists a connected graph $G$ with $src(G)=a$ and $src^\ell(G)=b$, and similarly for the pair $rc^\ell$ and $src^\ell$. Finally, we propose the question of whether or not we have $rc(G)=rc^\ell(G)$, for all connected graphs $G$.

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Theorems & Definitions (50)

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