Tight minimum colored degree condition for rainbow connectivity

TL;DR

This paper establishes a tight minimum colored degree condition in graphs that guarantees the existence of rainbow paths between any two vertices, extending previous results on properly-colored paths.

Contribution

It proves that a minimum colored degree of at least n/2 ensures rainbow connectivity between all vertex pairs, strengthening earlier bounds for rainbow paths.

Findings

Minimum colored degree condition guarantees rainbow connectivity

Extends previous properly-colored path results to rainbow paths

Provides tight bound for rainbow connectivity

Abstract

Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. In this paper, we show that the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path.