Every toroidal graph without triangles adjacent to $5$-cycles is DP-$4$-colorable

Abstract: DP-coloring, also known as correspondence coloring, is introduced by Dvo{\v{r}}{\'{a}}k and Postle. It is a generalization of list coloring. In this paper, we show that every connected toroidal graph without triangles adjacent to $5$-cycles has minimum degree at most three unless it is a 2-connected $4$-regular graph with Euler characteristic $\epsilon(G) = 0$. Consequently, every toroidal graph without triangles adjacent to $5$-cycles is DP-$4$-colorable. In the final, we show that every planar graph without two certain subgraphs is DP-$4$-colorable. As immediate consequences, (i) every planar graph without $3$-cycles adjacent to $4$-cycles is DP-$4$-colorable; (ii) every planar graph without $3$-cycles adjacent to $5$-cycles is DP-$4$-colorable; (iii) every planar graph without $4$-cycles adjacent to $5$-cycles is DP-$4$-colorable.