Abstract
Uniquely vertex colorable graphs and uniquely edge colorable graphs have been studied extensively by different authors. The literature on the similar problem for total coloring is void. In this paper we study this concept and, among other results, we prove that if a graph G ? K 2 is uniquely total colorable, then ??(G) = Δ + 1. Our results support the following conjecture: empty graphs, paths, and cycles of order 3k, k a natural number, are the only uniquely total colorable graphs.
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