Total Coloring Problem Of Embeddable Graphs Without Short Cycle

Graph theory is an important branch of discrete mathematics.Coloring theory is an important problem in graph theory.Coloring theory refers to coloring the vertices,edges or faces of a graph according to established rules,and dividing the vertices,edges or faces of a graph into corresponding subsets by different colors.Total coloring is one of the common coloring,and the problem of total coloring attract many scholars’ research and obtain many excellent results.A normal k total coloring of graph G =(V,E)refers to coloring the vertices and edges of the graph with k colors,so that any two adjacent elements of graph G can be assigned different colors.In this paper,we mainly study the total coloring of planar graphs and graphs that can be embedded in a surface of non negative Euler characteristic.Some existing results are optimized and promoted by our research.Specifically,this article is divided into four chapters for discussion.The first chapter mainly introduces the basic terms,relevant definitions,and symbols involved in this article,and provides a review of the current research status of total coloring problems.In addition,we briefly describe the key proof ideas used in this article and the main results presented.The second chapter mainly studies the total coloring problem of planar graphs and graphs that can be embedded in a surface of non negative Euler characteristic under restricted conditions,which proves the chromatic number of the graphs with Δ(G)≥ 8 is Δ(G)+ 1.The third chapter mainly proves that the total chromatic number is Δ(G)+ 1 of a graph that can be embedded in a surface of non negative Euler characteristic with Δ(G)≥ 7 and without some short cycles.These results generalize and improve some existing results.In chapter 4,we provide some perspectives on the total coloring problem of graphs and the application of coloring theory.