The Edge-Coloring of Graphs with Small Genus

Abstract

In this note, we prove that a graph is of class one if $G$ can be embedded in a surface with positive characteristic and satisfies one of the following conditions:(i) $\mathrm{\Delta }(G)\ge 3$ and $g(G)$(the girth of $G$) $\ge 8$ (ii) $\mathrm{\Delta }(G)\ge 4$ and $g(G)\ge 5$(iii) $\mathrm{\Delta }(G)\ge 5$ and $g(G)\ge 4$.