Equitable Total Chromatic Number of K× for p Even

A total coloring is equitable if the number of elements colored by any two distinct colors differs by at most one. The equitable total chromatic number of a graph ( χ e ″ ) is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that Δ + 1 ≤ χ e ″ ≤ Δ + 2 . In 1994, Fu proved that there exist equitable (Δ + 2)-total colorings for all complete r-partite p-balanced graphs of odd order. For the even case, he determined that χ e ″ ≤ Δ + 3 . Silva, Dantas and Sasaki (2018) verified Wang's conjecture when G is a complete r-partite p-balanced graph, showing that χ e ″ = Δ + 1 if G has odd order, and χ e ″ ≤ Δ + 2 if G has even order. In this work we improve this bound by showing that χ e ″ = Δ + 1 when G is a complete r-partite p-balanced graph with r ≥ 4 even and p even, and for r odd and p even.