Nordhaus-Gaddum-type theorem for rainbow connection number of graphs

TL;DR

This paper establishes bounds on the sum of rainbow connection numbers for a graph and its complement, providing sharp bounds for all graph sizes and expanding understanding of rainbow connectivity.

Contribution

It proves a Nordhaus-Gaddum-type theorem for rainbow connection numbers, giving sharp bounds for all graph sizes and characterizing extremal cases.

Findings

Lower bound of 4 for all n≥8, sharp for all n≥8.

Upper bound of n+2 for all n≥4, sharp for all n≥4.

Exact bounds for small n=4,5,6,7.

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 $G$, denoted $rc(G)$, is the minimum number of colors that are used to make $G$ rainbow connected. In this paper we give a Nordhaus-Gaddum-type result for the rainbow connection number. We prove that if $G$ and $\bar{G}$ are both connected, then $4\leq rc(G)+rc(\bar{G})\leq n+2$. Examples are given to show that the upper bound is sharp for all $n\geq 4$, and the lower bound is sharp for all $n\geq 8$. For the rest small $n=4,5,6,7,$ we also give the sharp bounds.