Spanning trees with many or few colors in edge-colored graphs

Abstract

Given a graph G = (V, E) and a (not necessarily proper) edgecoloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible.We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm.We use the minimum dominating set problem to show that the second problem is N P -hard.