The line graphs of Möbius ladder graphs are Type 1

Abstract

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 of a graph G is the smallest integer k for which G has a k-total coloring. If the total chromatic number of G is ∆(G) + 1, then we say that G is 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 the line graphs of Möbius ladder graphs, L(M2n), are Type 1.