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.
Keywords:
equitable total coloring; range coloring; regular graphs
