Dichromatic Number and Fractional Chromatic Number

The dichromatic number of a graph G is the maximum integer k such that there exists an orientation of the edges of G such that for every partition of the vertices into fewer than k parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and NeumannLara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We make the first significant progress on this conjecture by proving a fractional version of the conjecture. While our result uses a stronger assumption about the fractional chromatic number, it also gives a much stronger conclusion: if the fractional chromatic number of a graph is at least t , then the fractional version of the dichromatic number of the graph is at least 1 4 t/ log2(2et 2). This bound is best possible up to a small constant factor. Several related results of independent interest are given. 2010 Mathematics Subject Classification: 05C15