Two Sufficient Conditions for a Graph to be Type $1$

Abstract

In this paper, we give two sufficient conditions for a graph to be type $1$ with respect to the total chromatic number and prove the following results:

(i) If $G$ and $H$ are of type $1$, then $G\times H$ is of type $1$;

(ii) If $\epsilon (G)\le v(G)+\frac{3}{2}\mathrm{\Delta }(G)–4$, then $G$ is of type $1$.