Note on rainbow connection number of dense graphs

TL;DR

This paper investigates the rainbow connection number of dense graphs, establishing upper bounds based on minimum degree and diameter conditions, thus advancing understanding of coloring properties in dense graph classes.

Contribution

It provides new bounds on the rainbow connection number for dense graphs with specific degree and diameter conditions, extending prior work on rainbow connectivity.

Findings

For non-complete graphs with high minimum degree, rc(G) ≤ k.

Graphs with diameter 2 and high minimum degree have rc(G) ≤ k.

Certain bipartite graphs with common neighbors have rc(G) ≤ k.

Abstract

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. Following an idea of Caro et al., in this paper we also investigate the rainbow connection number of dense graphs. We show that for $k\geq 2$, if $G$ is a non-complete graph of order $n$ with minimum degree $\delta (G)\geq \frac{n}{2}-1+log_{k}{n}$, or minimum degree-sum $\sigma_{2}(G)\geq n-2+2log_{k}{n}$, then $rc(G)\leq k$; if $G$ is a graph of order $n$ with diameter 2 and $\delta (G)\geq 2(1+log_{\frac{k^{2}}{3k-2}}{k})log_{k}{n}$, then $rc(G)\leq k$. We also show that if $G$ is a non-complete bipartite graph of order $n$ and any two vertices in the same vertex class have at least $2log_{\frac{k^{2}}{3k-2}}{k}log_{k}{n}$ common neighbors in the other class, then $rc(G)\leq k$.