Coloring 3-power of 3-subdivision of subcubic graph
Fang Wang, Xiaoping Liu · Discrete Mathematics Algorithms and Applications · 2018
Let [Formula: see text] be a graph and [Formula: see text] be a positive integer. The [Formula: see text]-subdivision [Formula: see text] of [Formula: see text] is the graph obtained from [Formula: see text] by replacing each edge by a path of length [Formula: see text]. The [Formula: see text]-power [Formula: see text] of [Formula: see text] is the graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if the distance [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] is at most [Formula: see text]. Note that [Formula: see text] is the total graph [Formula: see text] of [Formula: see text]. The chromatic number [Formula: see text] of [Formula: see text] is the minimum integer [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-coloring. The total chromatic number of [Formula: see text], denoted by [Formula: see text], is the chromatic number of [Formula: see text]. Rosenfeld [On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc. 2 (1971) 405–408] showed that for a subcubic graph [Formula: see text], [Formula: see text]. In this note, we prove that for a subcubic graph [Formula: see text], [Formula: see text].