Rainbow vertex-connection and forbidden subgraphs

TL;DR

This paper characterizes specific graph families defined by forbidden subgraphs for which the rainbow vertex-connection number is tightly bounded by the graph's diameter plus a constant.

Contribution

It identifies all connected graph families with one or two forbidden subgraphs where the rainbow vertex-connection number is linearly bounded by diameter plus a constant.

Findings

Characterizes families with bounded rainbow vertex-connection number

Establishes bounds related to graph diameter for these families

Provides a complete classification for one- and two-forbidden subgraph cases

Abstract

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$, $rvc(G)\leq diam(G)+k_\mathcal{F}$, where $diam(G)$ is the diameter of $G$.