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Paper IPM / M / 2335  


Abstract:  
In this paper we investigate two conjectures proposed in (Graphs
Combin. 13(1997) 305314. The first one is uniquely totally
colorable(UTC) conjecture which states: Empty graphs, paths, and
cycles of order 3k, k a natural number, are the only UTC
graphs. We show that if G is a UTC graph of order n, then
∆ ≤ n / 2 + 1, where ∆ is the maximum
degree of G. Also there is another question about UTC graphs
that appeared in (Graphs Combin. 13 (1997) 305314) 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 vertax? 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 ∆regular graph and n/2 ≤ ∆ ≤ n/2 + 1.
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