Equitable total-coloring of subcubic graphs

An equitable total-coloring of a graph GG is a proper total-coloring such that the number of vertices and edges in any two color classes differ by at most one. Let χ″(G)χ″(G) and ΔΔ denote the total chromatic number and the maximum degree of a graph GG, respectively. In 1994, Fu conjectured that for any integer k≥max{χ″(G),Δ+2}k≥max{χ″(G),Δ+2}, GG is equitably total-kk-colorable. In this paper, we confirm this conjecture for the case Δ=3Δ=3.