Edge Irregular Total Labeling Problems Of Archimedean Tiling Graphs

For a graph G=(V,E),an edge irregular total k-labeling of G is a labeling (?):V?E?{1,2,…,k} such that the weights of any two different edges of G are distinct,where the weight of the edge xy under a total labeling (?) of G is defined as wt(xy)=(?)(x)+(?)(xy)+(?)(y).The total edge irregularity strength of G,denoted by tes(G),is the minimum k for which the graph G has an edge irregular total k-labeling.In 2006,Ivanc? and Jendrol' posed the following conjecture:for any graph G,which G is not isomorphic to K5,(?),where ?(G)is the maximum degree of a graph G.In this paper,we study edge irregular total labelings of subgraphs of three kinds of Archimedean tiling graphs,and determine the exact values of the total edge irregularity strength for these graphs and add further support to this conjecture.In Chapter 1,we introduce the research background and the main results of this thesis.In Chapter 2,we correct the error of Al-Mushayt et al.in proving the total edge irregularity strength of finite subgraphs of the tiling graph(63),and prove(?)by giving a correct labeling.Furthermore,we generalize the subgraph Hnm to a subgraph Hnm1,m2,…,mn of the tiling graph(63)and determine the exact value of the total edge irregularity strength of Hnm1,m2,…,mn.In Chapter 3,we study the edge irregular total labelings of finite subgraphs Tnm of the tiling graph(36),and prove(?).In Chapter 4.we prove(?)by constructing an edge irregular total(?)-labeling of finite subgraphs T Snm of the tiling graph(33.42).