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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 $3k, k $ a natural number, are the only UTC
graphs. We show that if $G$ is a UTC graph of order $n$, then
$\Delta \leq n / 2 + 1, $ where $\Delta $ 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 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 $\Delta$-regular graph and $n/2 \leq
\Delta \leq n/2 + 1$.
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