3-Rainbow index and forbidden subgraphs

TL;DR

This paper investigates the 3-rainbow index of graphs, establishing bounds related to forbidden subgraphs and Steiner diameters, and characterizes specific graph families with bounded 3-rainbow index.

Contribution

It characterizes finite graph families for which the 3-rainbow index is bounded by the 3-Steiner diameter plus a constant in connected, forbidden-subgraph-free graphs.

Findings

Identifies all finite families of forbidden subgraphs with bounded 3-rainbow index.

Establishes a relationship between 3-rainbow index and 3-Steiner diameter.

Provides bounds for the 3-rainbow index in terms of forbidden subgraphs.

Abstract

A tree in an edge-colored connected graph $G$ is called \emph{a rainbow tree} if no two edges of it are assigned the same color. For a vertex subset $S\subseteq V(G)$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$. A \emph{$k$-rainbow coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a rainbow $S$-tree in $G$. The minimum number of colors that are needed in a $k$-rainbow coloring of $G$ is the \emph{$k$-rainbow index} of $G$, denoted by $rx_k(G)$. The \emph{Steiner distance $d(S)$} of a set $S$ of vertices of $G$ is the minimum size of an $S$-tree $T$. The \emph{$k$-Steiner diameter $sdiam_k(G)$} of $G$ is defined as the maximum Steiner distance of $S$ among all sets $S$ with $k$ vertices of $G$. In this paper, we focus on the 3-rainbow index of graphs and find all finite families $\mathcal{F}$ of connected graphs, for which there is a constant $C_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$, $rx_3(G)\leq sdiam_3(G)+C_\mathcal{F}$.