Upper Bounding Rainbow Connection Number by Forest Number

TL;DR

This paper establishes a new upper bound on the rainbow connection number of a graph based on its forest number, revealing a stronger link between rainbow connectivity and forest structures than previously known.

Contribution

The paper proves that the rainbow connection number is at most the forest number plus two, improving understanding of rainbow coloring bounds in graphs.

Findings

Rainbow connection number is bounded by forest number plus two.

Counterexamples show bounds based on induced trees are insufficient.

New bound reveals a stronger relationship between rainbow connectivity and forest structures.

Abstract

A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph $G$ is the rainbow connection number of $G$, denoted by $\text{rc}(G)$. A simple way to rainbow-connect a graph $G$ is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of $G$. This proves that $\text{rc}(G) \le |V(G)|-1$. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of $t(G) -1$ for $\text{rc}(G)$, where $t(G)$ is the number of vertices in the largest induced tree of $G$? The answer turns out to be negative, as there are counter-examples that show that even $c\cdot t(G)$ is not an upper bound for $\text{rc}(G))$ for any given constant $c$. In this work we show that if we consider the forest number $f(G)$, the number of vertices in a maximum induced forest of $G$, instead of $t(G)$, then surprisingly we do get an upper bound. More specifically, we prove that $\text{rc}(G) \leq f(G) + 2$. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound.