Abstract:
A $D(2)$-vertex-distinguishing total coloring of a graph
$G$
is a proper total
coloring such that no pair of vertices, within distance two, has the same set of colors,
and the minimum number of colors required for such a coloring is called
$D(2)$-vertex-distinguishing total chromatic number of
$G$,
and denoted by
$\chi_{2vt}(G)$.
In this paper, we prove that
$\chi_{2vt}(G)\leq11$
for any graph
$G$
with
$\Delta(G)=3$.
Keywords:total coloring,
$D(2)$-vertex-distinguishing total coloring,
$D(2)$-vertex-distinguishing
total chromatic number.
