The Neighbor Expanded Sum Distinguishing Total Colorings Of Some Special Graphs

Let G be a simple graph.A total k-coloring of G is an assignment of k colors 1,2,…,k to all vertices and edges of G.For x ? V(G)and N(x)={y?V(G)|xy ? E(G)},the value(?)is called the expanded sum at x,where c is a total k-coloring of G.A total k-coloring c of G is called the neighbor expanded sum distinguishing(NESD for short)if w(x)?w(y),whenever xy ? E(G).The minimum number k of an NESD total k-coloring of G is called the neighbor expanded sum distinguishing total chromatic number of G and denoted by egndi ?(G).The neighbor expanded sum distinguishing total colorings of some special graphs are discussed in this paper by using the structure method and the method of mathematical induction,and prove that the neighbor expanded sum distinguishing total chromatic number of these special graph is less than or equal to 2.This result illustrates that the NESDTC conjecture proposed by Flandrin,et al is true for these special graphs.The following is the main framework of this paper.In chapter 1,the definition of neighbor sum distinguishing edge colorings and neighbor sum distinguishing total colorings of graph and related conclusions are introduced,further-more,the background and research significance of the neighbor expanded sum distinguishing total colorings of graph are presented.In chapter 2,the related concepts of the neighbor expanded sum distinguishing total colorings of graph are given,introduces the existing conclusions and conjecture of the neighbor expanded sum distinguishing total colorings of graph,the definition and related concepts of the whole article of special graphs are given.In chapter 3,the neighbor expanded sum distinguishing total colorings of some special graph are given,and get their the neighbor expanded sum distinguishing total chromatic number.In chapter 4,the neighbor expanded sum distinguishing total colorings of Several join of graphs are given,and get their the neighbor expanded sum distinguishing total chromatic number.In chapter 5,discuss the significance of the neighbor expanded sum distinguishing total colorings of these special graphs are given,and it is pointed out that research results of this paper provide ideas and methods for the follow up research.