Torus Graph DP-4-several Sufficient Conditions For Dyeability

Let G=(V,E)be a finite simple graph.We use V(G)and E(G)denote its vertex set and edge set,respectively.A list assignment of a graph G is a mapping that assigns each vertex v of G a set of colors L(v).If G has a coloring such that each vertex obtains a color in the list and the colors of two adjacent vertices are different,we say G is L-colorable.If G is L-colorable for every assignment L satisfying that|L(v)|≥k for all vertices v ∈ Y(G),then we say G is k-list-colorable or k-choosable.The list-chromatic number(or the choice number)of G,denoted by χl(G),is the minimum integer k such that G is k-choosable.Thomassen[20]showed that every planar graph is 5-choosable,and Voigt[21]showed that there are planar graphs which are not 4-choosable.Thus the 4choosability of planar graphs are studied by many scholars.List-coloring plays a very important role in the study of coloring of graphs,but some powerful tools in colorig are not feasiable for list-coloring.For example,the vertex identification technique.In order to overcome this difficulty,Dvorák and Postle[2]introduced the new concept of DP-coloring,which is a generalization of list-coloring.By using the new concept,they solved a long-standing conjecture of Borodin[22]on the list-coloring of planar graphs.They also proved that every planar graph is DP-5-colorable and the planar graphs with girth at least 5 are DP-3-colorable.The conjecture and problems of DPcoloring of planar graphs have been solved or proved by many researchers,so we are interested in whether the sufficient conditions of DP colorable of planar graphs can be extended to the toroidal graphs.In 2007,based on[6],Liu[14]proved that every toroidal graph without 4-cycles is 4-choosable.In 2009,Cai,Wang and Zhu[11]proved the choosability of toroidal graphs without short cycles:(?)In 2017,Kim and Ozeki[3]showed that for each k ∈ {3,4,5,6},every planar graph without k-cycles is DP-4-colorable.In 2019,Kim,Liu and Yu[4]showed that planar graphs without 7-cycles and butterflies are DP-4-colorable.In 2019,Li and Wang[7]showed that every totoidal graph without subgraphs isomorphic to the configurations in Fig 0-1 is DP-4-colorable.(?)In this thesis,we prove the following two results:(1)Toroidal graphs without 4-cycles are DP-4-colorable.(2)Toroidal graphs without 6,7-cycles are DP-4-colorable.