Total Coloring and Efficient Domination Applications to Non-Cayley Non-Shreier Vertex-transitive Graphs

Let . Let the star 2-set transposition graph be the -regular graph whose vertices are the -strings on symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such -string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph has the same vertex set of and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of . For , we show that and , among other intermediate transposition graphs, have total colorings via colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of and , and partitions into such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.