Equitable Total Chromatic Number of Kr×p 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 $\left({\chi }_e^{″}\right)$ is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that $\Delta +1\le {\chi }_e^{″}\le \Delta +2$. 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 ${\chi }_e^{″}\le \Delta +3$. Silva, Dantas and Sasaki (2018) verified Wang's conjecture when G is a complete r-partite p-balanced graph, showing that ${\chi }_e^{″}=\Delta +1$ if G has odd order, and ${\chi }_e^{″}\le \Delta +2$ if G has even order. In this work we improve this bound by showing that ${\chi }_e^{″}=\Delta +1$ when G is a complete r-partite p-balanced graph with r ≥ 4 even and p even, and for r odd and p even.