On the conformability of regular line graphs

Abstract

Let G = (V, E) be a graph and the deficiency of G be def(G) = ∑v∈V(G)(Δ(G)−dG(v)), where dG(v) is the degree of a vertex v in G. A vertex coloring φ:V(G)→{1,2,...,Δ(G)+1} is called conformable if the number of color classes (including empty color classes) of parity different from that of |V(G)| is at most def(G). A general characterization for conformable graphs is unknown. Conformability plays a key role in the total chromatic number theory. It is known that if G is Type 1, then G is conformable. In this paper, we prove that if G is k-regular and Class 1, then L(G) is conformable. As an application of this statement we establish that the line graph of complete graph L(Kn) is conformable, which is a positive evidence towards the Vignesh et al.’s conjecture that L(Kn) is Type 1.