Abstract: Let $G$ be a graph embedded on a surface $\Sigma$ of Euler characteristic $\chi(\Sigma)\!\geq\!0$.\\ $\chi'(G)$ and $\Delta(G)$ denote the chromatic index and the maximum degree of $G$, respectively. The paper shows that $\Delta(G)=\chi'(G)$ if the graph $G$ with $\Delta(G)\geq 4$ satisfies the following properties: (1) any two triangles with distance at least two; (2) any $i$-cycle and $j$-cycle with distance at least one, $i,j\in\{3,4\}$; (3) $G$ contains no $5$-cycles.

Key words: Euler characteristic, cycles, the chromatic index, discharging method