2-distance Sum Distinguishing Coloring Of Several Kinds Of Graphs

Graph theory is an important branch of mathematics,which is widely used.Graph coloring theory is an important part of graph theory,in which sum distinguishing coloring is one of the new research topics.Let ? be a proper edge coloring of graph G,for any u,v? V(G),if dG(u,v)?2 such that f(u)?f(v)where f(u)=?ux?E(G)?(ux),then ? is the 2-distance sum distinguishing edge coloring of graph G.The minimum number of the required colors of 2-distance sum distinguishing edge coloring for G is called the 2-distance sum distinguishing edge chromatic number,denote by x'2-?(G)for short.Let ? be a proper total coloring of graph G,for any u,v? V(G),if dG(u,v)?2 such that g(u)?g(v)where g(u)=?(u)+?u,x?E(G)?(ux),then ? is the 2-distance sum distinguishing total coloring of graph G.The minimum number of the required colors of 2-distance sum distinguishing total coloring for G is called the 2-distance sum distinguishing total chromatic number,denote by ?"2-?(G)for short.In this paper,we mainly study the 2-distance sum distinguishing edge coloring and total coloring of some graphs,and obtain an upper bound of the 2-distance sum distinguishing edge chromatic number and the 2-distance sum distinguishing total chromatic number of two kinds of graphs.There are five parts in this paper:In the first part,we mainly introduce the research background and the concepts and symbols involved in this paper.In the second part,we study the 2-distance sum distinguishing coloring of simple graphs such as path,circle,star,fan and wheel,and obtain their 2-distance sum dis-tinguishing edge chromatic number and 2-distance sum distinguishing total chromatic number.In the third part,We discuss the 2-distance sum distinguishing total chromatic num-ber of two classes of corona graphs,and obtain the corresponding chromatic numbers.In the fourth part,we first prove the 2-distance sum distinguishing edge chromatic number of the tree graph,and then consider the relationship between the tree and Halin graph,we get a upper bound of 2-distance sum distinguishing edge chromatic number of 3-regular Halin graph by the Combinatorial Nullstellensatz.Through the induction and classification of the number of interior points of 3-regular Halin graphs,we also get a upper bound of 2-distance sum distinguishing total chromatic number of 3-regular Halin graph by the Combinatorial Nullstellensatz.In the fifth part,by analyzing the structure of outerplanar graphs,we study the 2-distance sum distinguishing coloring of outerplanar graphs,and give an upper bound of the 2-distance sum distinguishing edge chromatic number of outerplanar graphs,and an upper bound of the 2-distance sum distinguishing total chromatic number of outerplanar graphs.