Fractional total colourings of graphs of high girth

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