A Characterization of Graphs with Fractional Total Chromatic Number Equal to

For a simple graph of maximum degree Δ, the complexity of computing the fractional total chromatic number is unknown. Trivially it is at least Δ + 1 . Kilakos and Reed proved that it is at most Δ + 2 . In this paper, we strengthen this by characterizing exactly those simple graphs with fractional total chromatic number Δ + 2 . This yields a simple linear-time algorithm to determine whether a given graph has fractional chromatic number Δ + 2 .