An Extremal Problem on Rainbow Spanning Trees in Graphs

TL;DR

This paper investigates the extremal counts of rainbow spanning trees in edge-colored graphs, providing exact bounds for specific classes and general graphs, and introduces new counting methods.

Contribution

It offers complete solutions for counting rainbow spanning trees in JL-colored complete and bipartite graphs, and develops an analogue of Kirchhoff's theorem for general graphs.

Findings

Tight bounds for rainbow spanning trees in complete graphs and bipartite graphs.

Characterization of graphs with minimal rainbow spanning trees under JL-colorings.

An analogue of Kirchhoff's matrix tree theorem for counting rainbow spanning trees.

Abstract

A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and complete bipartite graphs $K_{n,m}$. In both cases, we find tight upper and lower bounds; the lower bound for $K_n$, in particular, proves to have an unexpectedly chaotic and interesting behavior. We further investigate this question for JL-colorings of general graphs and prove several results including characterizing graphs which have JL-colorings achieving the lowest possible number of rainbow spanning trees. We establish other results for general $n-1$ colorings, including providing an analogue of Kirchoff's matrix tree theorem which yields a way of counting rainbow spanning trees in a general graph $G$.