A Note on The \((g, f)\)-Chromatic Index of Graphs

Abstract

A \((g, f)\)-coloring is a generalized edge-coloring in which each color appears at each vertex \(v\) at least \(g(v)\) and at most \(f(v)\) times, where \(g(v)\) and \(f(v)\) are nonnegative and positive integers assigned to each vertex \(v\), respectively. The minimum number of colors used by a \((g, f)\)-coloring of \(G\) is called the \((g, f)\)-chromatic index of \(G\). The maximum number of colors used by a \((g, f)\)-coloring of \(G\) is called the upper \((g, f)\)-chromatic index of \(G\). In this paper, we determine the \((g, f)\)-chromatic index and the upper \((g, f)\)-chromatic index in some cases.