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 \(\varepsilon(G) \leq v(G) + \frac{3}{2}\Delta(G) – 4\), then \(G\) is of type \(1\).