2-Distance Sum Distinguishing Coloring Of Some Classes Of Graphs

Let G=G（V,E）be a connected simple graph with order not less than 3.A proper edge coloring f of graph G is called 2-distance sum distinguishing edge coloring（D（2）VSDEC for short）if for any u,v ∈ V（G）,d_G（u,v）≤2 such that S（u）≠S（v）,where S（u）=∑_{uω∈E（G）}f（uω）.The 2-distance sum distinguishing edge chromatic numbers of graph G is the smallest integer k such that the graph G has a 2-distance sum distinguishing edge coloring,denote by χ_2-∑’（G）for short.A proper total coloring Φ of graph G is called 2-distance sum distinguishing total coloring（D（2）-VSDTC for short）if for any u,v ∈ V（G）,d_G（u,v）≤2 such that g（u）≠g（v）7 where g（u）=Φ（u）+∑_{uω∈E（G）}Φ（uω）.The 2-distance sum distinguishing total chromatic numbers of graph G is the smallest integer k such that the graph G has a 2-distance sum distinguishing total coloring,denote by x_2-∑"（G）for short.In this paper,we mainly study the 2-distance sum distinguishing edge（total）coloring of three special graphs by using mathematical induction、constructing coloring function、Combinatorial Nullstellensatz and Discharging method,and obtain their 2-distance sum distinguishing edge chromatic numbers and 2-distance sum distinguishing total chromatic numbers.This thesis mainly consists of four chapters:In chapter 1,we mainly introduce the research background of the problem of vertex sum distinguishing coloring of graphs and the concepts and symbols involved in this paper.In chapter 2,we study the 2-distance sum distinguishing coloring of tree graphs,and obtain their 2-distance sum distinguishing edge（total）chromatic numbers.In chapter 3,we study the 2-distance sum distinguishing coloring of unicyclic graphs,and obtain their 2-distance sum distinguishing edge（total）chromatic numbers.In chapter 4,we study the 2-distance sum distinguishing coloring of bicyclic graphs,and obtain their 2-distance sum distinguishing edge（total）chromatic numbers.