Fractional total colourings of graphs of high girth

Abstract

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1.