Some Results on Fractional Edge Coloring of Graphs

Abstract

A fractional edge coloring of graph \(G\) is an assignment of a nonnegative weight \(w_M\) to each matching \(M\) of \(G\) such that for each edge \(e\) we have \(\sum_{M\ni e} w_M \geq 1\). The fractional edge coloring chromatic number of a graph \(G\), denoted by \(\chi’_f(G)\), is the minimum value of \(\sum_{M} w_M\) (where the minimum is over all fractional edge colorings \(w\)). It is known that for any simple graph \(G\) with maximum degree \(\Delta\), \(\Delta < \chi'_f(G) \leq \Delta+1\). And \(\chi'_f(G) = \Delta+1\) if and only if \(G\) is \(K_{2n+1}\). In this paper, we give some sufficient conditions for a graph \(G\) to have \(\chi'_f(G) = \Delta\). Furthermore, we show that the results in this paper are the best possible.