Some results on fractional edge coloring of graphs

A fractional edge coloring of graph G is an assignment of a nonnegative weight ω_M to each matching M of G such that for each edge e we have Σ_M∋eω_M ≥ 1. The fractional edge coloring chromatic number of a graph G, denoted by ξ′_f(G), is the minimum value of Σ_M ω_M(where the minimum is over all fractional edge colorings ω). It is known that for any simple graph G with maximum degree , Δ, Δ ≤ χ′ _f(G) ≤ Δ + 1. And χ′_f/(G) = A + 1 if and only if G is K_2n+1. In this paper, we give some sufficient conditions for a graph G to have χ′_f(G) = Δ. Furthermore we show that the results in this paper is the best possible.