Some results on an edge coloring problem of Folkman and Fulkerson Original Research Article

In 1968, Folkman and Fulkerson posed the following problem: Let G be a graph and let (n1,…,nt) be a sequence of positive integers. Does there exist a proper edge coloring of G with colors 1,2,…,t such that precisely ni edges receive color i, for each i=1,…,t? If such a coloring exists then the sequence (n1,…,nt) is called color-feasible for G. Some sufficient conditions for a sequence to be color-feasible for a bipartite graph where found by Folkman and Fulkerson, and de Werra. In this paper we give a generalization of their results for bipartite graphs. Furthermore, we find a set of color-feasible sequences for an arbitrary simple graph. In particular, we describe the set of all sequences which are color-feasible for a connected simple graph G with Δ(G)⩾3, where every pair of vertices of degree at least 3 are non-adjacent.