Note on minimally $k$-rainbow connected graphs

TL;DR

This paper improves lower bounds on the minimum number of edges needed for $k$-rainbow connected graphs, specifically providing new bounds for cases where the diameter is between 3 and half of the number of vertices.

Contribution

It advances the understanding of rainbow connectivity by establishing new lower bounds for the minimum size of $k$-rainbow connected graphs for certain diameters.

Findings

Improved lower bound for $t(n,2)$.

New lower bounds for $t(n,d)$ when $3 \\leq d < \\lceil n/2 \\rceil$.

Abstract

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil $. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.