More on rainbow disconnection in graphs

TL;DR

This paper investigates the rainbow disconnection number in graphs, solving a conjecture about maximum size for given parameters, establishing bounds for various graph classes, and deriving a Nordhaus-Gaddum-type theorem relating a graph and its complement.

Contribution

It solves a conjecture on maximum size of graphs with fixed rainbow disconnection number and establishes bounds and exact values for special graph classes, including a Nordhaus-Gaddum-type theorem.

Findings

Solved the maximum size conjecture for graphs with given rainbow disconnection number.

Established bounds for rainbow disconnection numbers in complete multipartite, critical, minimal, and regular graphs.

Proved a Nordhaus-Gaddum-type theorem relating the rainbow disconnection numbers of a graph and its complement.

Abstract

Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there exists a $u-v$ rainbow cut. For a connected graph $G$, the rainbow disconnection number of $G$, denoted by $rd(G)$, is defined as the smallest number of colors that are needed in order to make $G$ rainbow disconnected. In this paper, we first solve a conjecture that determines the maximum size of a connected graph $G$ of order $n$ with $rd(G) = k$ for given integers $k$ and $n$ with $1\leq k\leq n-1$, where $n$ is odd, posed by Chartrand et al. in \cite{CDHHZ}. Secondly, we discuss bounds of the rainbow disconnection numbers for complete multipartite graphs, critical graphs, minimal graphs with respect to chromatic index and regular graphs, and give the rainbow disconnection numbers for several special graphs. Finally, we get the Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We prove that if $G$ and $\overline{G}$ are both connected, then $n-2 \leq rd(G)+rd(\overline{G})\leq 2n-5$ and $n-3\leq rd(G)\cdot rd(\overline{G})\leq (n-2)(n-3)$. Furthermore, examples are given to show that the upper bounds are sharp for $n\geq 6$, and the lower bounds are sharp when $G=\overline{G}=P_4$.