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A totally efficient coloring of a regular graph with color set formed by one unit more than its degree is a total coloring in which each vertex color class is an efficient dominating set, that is a perfect code. We find that the 3-cube graph has a totally efficient coloring and conjecture that this is the only existing case of a totally effective color We also deal with a related problem exemplified by star transposition graphs and related graphs, leading to edge-girth colorings on their prism graphs.
Italo J. Dejter (Tue,) studied this question.
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