| Let G=（V（G）,E（G））be a simple connected graph with|V|=n and|E|=m.A bijection f:E（G）→{1,2,...,m}is called a local antimagic labeling if for any two adjacent vertices u and v,ω（u）=ω（v）,whereω（u）=e∈E（u）f（e）,and E（u）is the set of edges incident to u.Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the colorω（v）.The local antimagic chromatic numberχla（G）is the minimum number of colors taken over all colorings induced by local antimagic labelings of G.Let G=（V（G）,E（G））be a graph of vertex set V（G）and edge set E（G）.Local antimagic total coloring developed from local edge and local vertex antimagic coloring of graph.Local antimagic total coloring is defined as f:E（G）∪V（G）→{1,2,...,|V（G）|+|E（G）|}if for any two adjacent vertices u1and u2,ωt（u1）≠ωt（u2）,where for u∈V（G）,ωt（u）=e∈E（u）φ（e）+φ（u）,where E（u）is respectively the set of edges incident to u.Thus,any local antimagic total labeling induces a proper vertex labeling of G if each vertex u is assigned the colorω（u）.The chromatic number of local antimagic total labeling denoteχlat（G）is the minimum number of colors taken over all colorings induced by local antimagic total labeling of graph G.In this paper it mainly studies the influence of graphs operations and graphs substructures on local antimagic chromatic numbers,and then presents the calculation of local antimagic（total）chromatic number of some composite graphs.This paper is divided into four chapters.The first chapter mainly introduce the development background and advance of the local antimagic（total）labeling,as well as the basic definitions and concepts covered in this paper.Then,listing some main results in this paper.In the second chapter,Arumugam et al.gave the local antimagic chromatic number of friend-ship graph after deleting an arbitrary edge,while this chapter consideres the influence of friendship graph with adding a pendant edge on the local antimagic chromatic number.In the third chapter,for any graph G,Lau et al.studied the relationship between the local antimagic chromatic number of graph G after edge deletion and edge addition,and the local an-timagic chromatic number of the original graph G.This chapter considers the local antimagic chromatic number of subdivivion graphs of star graph、double star graph、triangle book graph、quadrangle book graph and octopus graph,as the same time,studying the local antimagic chro-matic number of blossomed star.The fourth chapter studies the local antimagic total chromatic of flowerpot graph. |
