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

03/03/2018

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by   PetsTime, et al.

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DP-coloring, also known as correspondence coloring, is introduced by Dvořák 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 ϵ(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.