Equitable total-coloring of subcubic graphs

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