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 $\sum _{M\ni e}{w}_M\ge 1$. The fractional edge coloring chromatic number of a graph $G$, denoted by ${\chi }_f^{\prime }(G)$, 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 $\mathrm{\Delta }$, $\mathrm{\Delta }<{\chi }_f^{\prime }(G)\le \mathrm{\Delta }+1$. And ${\chi }_f^{\prime }(G)=\mathrm{\Delta }+1$ if and only if $G$ is $K_{2n+1}$. In this paper, we give some sufficient conditions for a graph $G$ to have ${\chi }_f^{\prime }(G)=\mathrm{\Delta }$. Furthermore, we show that the results in this paper are the best possible.