DP-4-colorable Graphs On The Torus

DP-coloring was proposed by Dvorak and Postle,and is a generalization of list-coloring.Given every vertex a list assignment L,L:v ? G?L(v).Define Lv={v}×L(v),ML?uv be a matching between Lu and Lv for any edge uv ? E(G),ML={ML,uv:uv ? E(G)} is a matching assignment of G over L.In particular,GL is a ML-cover of G,if GL meets the following four conditions.(1)The vertex set of GL is Uv?V(G)Lv={(v,c):u ?V(G),c ?L(v)}.(2)Graph GL[Lv]is a clique for any v ?V(G).(3)if uv ? E(G),the edges between Lu and Lv are all in ML,uv.(4)if uv(?)E(G),there is no any edge between Lu and Lv.And if ML-cover has an independent set I,|I|=|V(G)| and |I?L(v)|=1,then I is a ML-coloring of G.In particular,if G has a ML-coloring for any k-list assignment L and matching assignment ML over L,then G is DP-k-colorable,and the minimum positive integer k that makes graph G be DP-k-colorable is called DP chromatic number,denoted by ?DP(G).In 2018 and 2019,Kim,Ozeki and Yu showed that:Theorem 1.[5]Every planar graph without 4-cycles is DP-4-colorable.Theorem 2.[6]Every planar graph without 3-cycles adjacent to 4-cycles is DP-4-colorable.In this paper,we extend the above two results to torus,and get the following results:Theorem 3.Let G be a toroidal graph.(1)If G does not contain 4-cycles,then G is DP-4-colorable.(2)If 3-cycles of G is not adjacent to 4-cycles,then G is DP-4-colorable.