On the conformability of regular line graphs

Abstract

Let $G=(V,E)$ be a graph and the \emph{deficiency of $G$}  be $def(G)=\sum_{v \in V(G)} (\Delta(G)-d_{G}(v))$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A vertex coloring $\varphi :V(G)\to \{1,2,...,\Delta(G)+1\}$ is called \emph{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 \textit{Type~1}, then $G$ is conformable. In this paper, we prove that if $G$ is $k$-regular and \textit{Class~1}, then $L(G)$ is conformable. As an application of this statement we establish that the line graph of complete graph $L(K_n)$ is conformable, which is a positive evidence towards the Vignesh et al.'s  conjecture that $L(K_n)$ is \textit{Type~1}.