A Characterization of Graphs with Fractional Total Chromatic Number Equal to Δ+2

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.