Total coloring line graphs of generalized Petersen graphs

Resumo

A k-total coloring of G is an assignment of k colors to its elements (vertices and edges) such that adjacent or incident elements have distinct colors. The total chromatic number is the smallest integer k for which the graph G has a k-total coloring. If the total chromatic number is ∆(G)+1, then G is called Type 1. The line graph of G, denoted by L(G), is the graph whose vertex set is the edge set of G and two vertices of the line graph of G are adjacent if the corresponding edges are adjacent in G. In this paper, we prove that every line graph of the generalized Petersen graph is conformable; the graph L(G(n, 1)), for n ≥ 3, is Type 1; and, if L(G(n, k)) is Type 1, then L(G(n′, k′)) is Type 1, for all n′ ≡ 0 mod n and k′ ≡ k mod n.