Two conjectures on uniquely totally colorable graphs

Abstract

In this paper we investigate two conjectures proposed in (Graphs Combin. 13 (1997) 305–314). The first one is uniquely totally colorable (UTC) conjecture which states: Empty graphs, paths, and cycles of order 3 k, k a natural number, are the only UTC graphs. We show that if G is a UTC graph of order n, then Δ⩽ n/2+1, where Δ is the maximum degree of G. Also there is another question about UTC graphs that appeared in (Graphs Combin. 13 (1997) 305–314) as follows: If a graph G is UTC, is it true that in the proper total coloring of G, each color is used for at least one vertex? We prove that if G is a UTC graph of order n and in the proper total coloring of G, there exists a color which did not appear in any vertex of G, then G is a Δ-regular graph and n/2⩽ Δ⩽ n/2+1.