On the conformability of regular line graphs

Let G = (V, E) be a graph and the deficiency of G be def(G) = n-ary sumation v is an element of V(G)(Delta(G)-dG(v))def(G) = n-ary sumation v is an element of V (G)(Delta(G)-dG(v))$ {def}(G)\enspace =\enspace {\sum }_{v\in V\enspace (G)}<^>{}(\Delta (G)-{d}_G(v))$, where dG(v) is the degree of a vertex v in G. A vertex coloring phi:V(G)->{1,2,...,Delta(G)+1}phi: V (G) -> {1, 2, . . ., Delta(G) + 1}$ \phi:\enspace V\enspace (G)\enspace \to \enspace \{1,\enspace 2,\enspace.\enspace.\enspace.,\enspace \Delta (G)\enspace +\enspace 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.