Anti-Ramsey number of edge-disjoint rainbow spanning trees

TL;DR

This paper proves a conjecture about the maximum number of colors in an edge-colored complete graph that avoids having t edge-disjoint rainbow spanning trees, providing a complete characterization of this anti-Ramsey number.

Contribution

The paper confirms the conjecture by Jahanbekam and West and determines the anti-Ramsey number for all values of n and t, including the case when n=2t+1.

Findings

Proved the conjecture for n ≥ 2t+2.

Determined the anti-Ramsey number for n=2t+1.

Complete characterization of the anti-Ramsey number for all n and t.

Abstract

An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$.