An Extremal Problem on Rainbow Spanning Trees in Graphs

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.