Degree Matrices Realized by Edge-Colored Forests

Given a c-edge-colored graph G on n vertices, we define the degree matrix M as the c× n matrix whose entry dij is the degree of color i at vertex vj . We show that the obvious necessary conditions for a c× n matrix to be the degree matrix of a c-edge-colored forest on n vertices are also sufficient. It is well known that non-negative integers d1, d2, . . . , dn form a degree list of a forest on n vertices if and only if ∑ di is even and ∑ di ≤ 2s − 2 where s is the number of non-zero entries in d1, d2, . . . , dn. We are interested in similar conditions for edge colored forests where we specify the number of incident edges of each color. Given an edge colored forest and any set of colors, the edges of those colors induce a forest. Thus the degree sums for these colors must satisfy the conditions for uncolored forests. We will show that this necessary condition is also sufficient. The three color version of our problem is related to results in [1] and [2]. In those papers, two of the three colors each induce a forest (but not necessarily their union) and the third color is the complement of their union. For our results, if every entry is 0, 1 or 2 then the forest would be a disjoint union of paths. If we further specify the lengths of these paths, conditions for a realization become more complicated. We will examine these in a forthcoming paper. We start by formalizing our notation. Definition 1. Given a graph G, a c edge coloring is an assignment c : EG → [c] from its edge set into the set [c] = {1, 2, . . . , c} whose elements are called colors. For any c, such an assignment is called an edge-coloring. If G is assigned such a coloring, then G is called an edge-colored graph.