RESUMO Neste trabalho introduzimos os conceitos de coloração total absolutamente equilibrada e composição de grafos. Provamos que para n , K ∈ ℕ, se k + 1 | n, existe um grafo k-regular conexo com n vértices que admite uma coloração total absolutamente equilibrada com no máximo Δ + 2 cores. Esse resultado mostra que existe uma relação entre a regularidade e o número de vértices do grafo que possibilita a construção de uma família de grafos regulares, denominados grafos harmônicos. Em seguida, mostramos que todo grafo harmônico de grau k pode ser obtido como composição sucessiva de grafos completos de grau k. Finalizamos, provando que os grafos harmônicos não possuem vértice de corte, fato que implica que todo grafo desta família possui conectividade de vértices κ G ≥ 2.

ABSTRACT In this work we introduce the concepts of absolute equitable total coloring and graph composition. We prove that for n , K ∈ ℕ, if k + 1 | 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 κ G ≥ 2.

ABSTRACT This work aims to study the equitable total coloring into subfamilies of regular graphs. For this purpose, we use some relationships between equitable total coloring and range (vertex) coloring in some regular graphs. The concept of range coloring of order k was first presented by (Lozano et al., 2009). In this paper, we shows that if a regular graph G admits an equitable range coloring c of order Δ with (Δ+1) colors then there is an equitable total coloring of G - with the same set of colors - that extends c. We also show that there are infinite graphs satisfying this theorem. Such graphs are called Harmonics. We generate Harmonic Graphs which are Cartesian products of cycles and their complements. These graphs are regular and they admit an equitable total coloring under the above conditions.