Coloração Total Absolutamente Equilibrada em uma Família de Grafos Regulares

ABSTRACT

In this work we introduce the concepts of absolute equitable total coloring and graph composition. We prove that for $n,K\in \mathrm{\mathbb{N}}$, if $\left(k+1\right)|n$, there is a connected k regular graph with n vertices that admits a absolute equitable total coloring, with at most Delta + 2 colors. This result shows that there is a relationship between regularity and the number of vertices of the graph that makes it possible to build a family of regular graphs, called harmonic graphs. Then, we show that every harmonic graph of degree k can be obtained as successive composition of complete graphs of degree k. We conclude by proving that the harmonic graphs do not have a cut vertex, that implies that every graph of this family has vertex connectivity $\kappa \left(G\right)\ge 2$.

Keywords:
harmonic graphs; graph composition; vertex connectivity