k-DEPENDENCE, DISJOINT MATCHINGS, AND AN EXTENSION OF A THEOREM OF FAVARON

A vertex set D in a graph G is k-dependent if G[D] has maximum degree at most k−1, and k-dominating if every vertex outside D has at least k neighbors in D. Favaron [2] proved that if D is a k-dependent set maximizing the quantity k |D|−|E(G[D])|, then D is k-dominating. We extend this result, showing that such sets satisfy a stronger property: given any ordering < of V (G)−D, there is a k-edge-chromatic subgraph of G in which every vertex v outside D has degree at least k− d−(v), where d−(v) is the number of earlier neighbors of v in V (G) − D. Since any vertex outside D may be taken as a minimal element of <, this implies that D is k-dominating.