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 ?? with (??+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.
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机译：这项工作的目的是研究将均匀的总着色划分为正则图的子族。为此，我们在某些规则图中使用合理的总着色与范围（顶点）着色之间的某些关系。 （Lozano et al。，2009）首次提出了k阶范围着色的概念。在本文中，我们表明，如果正则图G允许等距范围c的着色c？具有（?? + 1）种颜色，则G具有相等的全部颜色-具有相同的颜色-扩展了c。我们还表明，存在满足该定理的无限图。这样的图称为谐波。我们生成谐波图，它是周期及其补的笛卡尔积。这些图是规则的，并且在上述条件下它们允许相等的总着色。
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