Graphs that embedded in any fixed surfaces with sufficiently large maximum degree $ \Delta $ is total-$ (\Delta +1) $-colorable

Abstract

<abstract><p>Let $ G $ be a graph that can be embedded in a surface $ \Sigma $ of Euler characteristic $ c'(\Sigma) $. In this paper, we proved that there exists an integer $ \Delta_0 = 4\cdot (5+\sqrt{49-24c'})\cdot [2-c'+\frac{1}{2} (5+\sqrt{49-24c'})] $ such that the total chromatic number of $ G $ is $ \Delta(G) +1 $ if $ \Delta(G) \geq \Delta_0 $.</p></abstract>