Maximal edge colorings of graphs

Abstract

For a graph G of order n a maximal edge coloring is a proper edge coloring with χ′(Kn) colors such that adding any edge to G in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges there are no graphs with a maximal edge coloring, while for some other values, they provided constructions of such graphs. However, for many values of the number of edges determining whether there exists any graph with a maximal edge coloring remained open. Here, we solve the problem completely by determining for any number of vertices and edges if there exists a graph with a maximal edge coloring.