A note on the minimum size of $k$-rainbow connected graphs

TL;DR

This paper determines the exact minimum number of edges needed for a graph to be k-rainbow connected, providing a precise formula for all sufficiently large graphs and color counts.

Contribution

It establishes a closed-form formula for the minimum size of k-rainbow connected graphs for all n, k ≥ 3, advancing understanding of rainbow connectivity thresholds.

Findings

Derived the exact formula t(n,k) = ⌈k(n-2)/(k-1)⌉ for all n, k ≥ 3.

Confirmed the formula's validity for all large enough graphs and color counts.

Enhanced theoretical understanding of rainbow connectivity in graph theory.

Abstract

An edge-coloured graph $G$ is rainbow connected if there exists a rainbow path between any two vertices. A graph $G$ is said to be $k$-rainbow connected if there exists an edge-colouring of $G$ with at most $k$ colours that is rainbow connected. For integers $n$ and $k$, let $t(n,k)$ denote the minimum number of edges in $k$-rainbow connected graphs of order $n$. In this note, we prove that $t(n,k) = \lceil k(n-2)/(k-1) \rceil$ for all $n, k \ge 3$