Maximal edge colorings of graphs

TL;DR

This paper completely characterizes the existence of graphs with maximal edge colorings, resolving open questions about which edge counts admit such graphs.

Contribution

Provides a complete solution to the problem of determining for which edge counts graphs with maximal edge colorings exist.

Findings

Identifies all edge counts for which maximal edge colorings exist.

Constructs examples of graphs with maximal edge colorings for certain edge counts.

Proves non-existence for other edge counts.

Abstract

For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $\chi'(K_n)$ 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. We give a complete solution of this problem.