Coloring 3-Power Of 3-Subdivision Of Subcubic Graph

Let G be a graph and k be a positive integer. The k-subdivision S-k(G) of G is the graph obtained from G by replacing each edge by a path of length k. The k-power G(k) of G is the graph with vertex set V (G) in which two vertices u and v are adjacent if and only if the distance d(G)(u, v) between u and v in G is at most k. Note that S-2(G)(2) is the total graph T(G) of G. The chromatic number chi(G) of G is the minimum integer l for which G has a proper l-coloring. The total chromatic number of G, denoted by chi ''(G), is the chromatic number of T(G). 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 G, chi ''(G) <= 5. In this note, we prove that for a subcubic graph G, chi(S-3(G)(3)) <= 7.