Total colorings of k-regular graphs of girths 2k and k

Let \ (2 k\). We say that a total coloring of a \ (k\) -regular simple graph via \ (k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. We prove that Hamming shells of star transposition graphs and Hamming cubes have efficient total colorings. Also in this work, a focus is set upon the graphs of girth \ (2k\) and \ (k\). Efficient total colorings of finite connected simple cubic graphs of girth 6 are constructed. These are of two specific types, namely: (a) those whose 6-cycles use just 3 colors with antipodal monochromatic pairs of vertices or edges; (b) those whose 6-cycles do not respect item (a) so they use four colors. An orthogonality property holds for all graphs of type (a). Such property allows further edge-half-girth colorings in the corresponding prism graphs.