A New Sufficient Condition for Graphs of \(f\)-Class \(1\)

Abstract

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\). A simple graph \(G\) is of \(f\)-class 1 if the \(f\)-chromatic index of \(G\) equals \(\Delta_f(G)\), where \(\Delta_f(G) = \max_{v\in V(G)}\{\left\lceil\frac{d(v)}{f(v)}\right\rceil\}\). In this article, we find a new sufficient condition for a simple graph to be of \(f\)-class 1, which is strictly better than a condition presented by Zhang and Liu in 2008 and is sharp. Combining the previous conclusions with this new condition, we improve a result of Zhang and Liu in 2007.