Graphs of f-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v∈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 Δf(G), where Δf(G)=maxv∈V(G){⌈dG(v)∕f(v)⌉}. 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 et. al (2010). As a consequence, this result extends earlier results of Hakimi and Schmeichel, Hoffman and Rodger, Akbari, Cariolaro, Chavooshi, Ghanbari and Zare on class 1 graphs.