Research On The Local Antimagic (Total) Chromatic Numbers Of Several Composition Graphs

Let G=(V(G),E(G))be a connected simple graph with n vertices and m edges.A bijection f:E(G)?{1,2,…,m} is called local antimagic labeling,if for any two adjacent vertices u and v of G we have ?(u)?(v).where ?(u)=?e?E(u)f(e),and E(u)is the set of edges incident to u.Any local antimagic labeling of graph G induces a proper vertex coloring of G if the vertex v is assigned the color ?(v).The local antimagic chromatic number of graph G,denoted by ?la(G),is the minimum number of colors taken over all colorings induced by local antimagic labeling of G.A bijection g:V(G)? E(G)? {1,2,…,n+m} is called local antimagic total labeling,if for any two adjacent vertices u and v of G satisfying with ?t(u)??t(v),where ?t(u)=g(u)+?e?E(u)g(e).Any local antimagic total labeling of graph G induces a proper vertex coloring of G if the vertex v is assigned the color ?t(v).The local antimagic total chromatic number of graph G,denoted by ?lat(G),is the minimum number of colors taken over all colorings induced by local antimagic total labeling of G.The paper mainly studies some results on the local antimagic(total)chromatic numbers of several composition graphs.This paper is divided into four chapters.The first chapter mainly introduces the background and advance of the local antimagic(total)coloring problems.Basic definitions and concepts are covered in this paper.Then the first chapter presents some main results in this paper.The second chapter studies the exact values of the local antimagic chromatic numbers on the corona product of graph G and empty graph Km,where G is the friendship graph Fn or the star Sn.The third chapter determines the local antimagic chromatic numbers of the join graphs Fn ?K2,Pn ? K2 and the join graphs Pn ? K2?Cn ? K2?Sn ? K2 and Fn ? K2.The fourth chapter presents the local antimagic total chromatic numbers of some wheelrelated graphs,such as the fan graph Fn,the bowknot graph Bn,n,the Dutch windmill graph D4n,the analogous Dutch windmill graph AD4n and the flower graph Fn.