Data: 28/11/2018

Título: Equitable total chromatic number of Kr×p for p even

Palestrante: Anderson Gomes da Silva, PUC/RJ
Data: 28 de novembro de 2018, 9 h.
Local: Sala 407, Bloco H, Campus Gragoatá, UFF.

Resumo:  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 (\(\chi''_e\)) is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that \(\Delta+1\leq\chi''_e\leq\Delta+2\). In 1994, Fu proved that there exist equitable ($\Delta+2$)-total colorings for all complete $r$-partite $p$-balanced graphs of odd order. For the even case, he determined that $\chi''_e\leq\Delta+3$.

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 \(\chi''_e\leq\Delta+2\) 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 \(r\geq4\) even and \(p\) even, and for \(r\) odd and \(p\) even.