Equitable Total Chromatic Number of Kr×p for p even

Abstract

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 (χe″) is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that Δ+1≤χe″≤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 χe″≤Δ+3. Silva, Dantas and Sasaki (2018) verified Wang's conjecture when G is a complete r-partite p-balanced graph, showing that χe″=Δ+1 if G has odd order, and χe″≤Δ+2 if G has even order. In this work we improve this bound by showing that χe″=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.