Anti-Ramsey number of edge-disjoint rainbow spanning trees in all graphs

TL;DR

This paper determines the anti-Ramsey number for the maximum count of colors in edge-colorings of any graph that avoid having t edge-disjoint rainbow spanning trees, providing explicit formulas for all graphs and specific cases like complete bipartite graphs.

Contribution

It provides a complete characterization of the anti-Ramsey number for all graphs and all values of t, extending previous results specifically for complete bipartite graphs.

Findings

Explicit formulas for r(G,t) for all graphs G

Determination of r(K_{p,q},t) for all p, q, t

Improved bounds and exact values compared to prior work

Abstract

An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the maximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|<t(|P_0|-1)$; and $r(G,t)=\max_{P\colon |P|\geq 3} \{|E(P,G)|+t(|P|-2)\}$ otherwise. As a corollary, we determine $r(K_{p,q},t)$ for all values of $p,q, t$, improving a result of Jia, Lu and Zhang.