Total Vertex Irregularity Strength Of Certain Equitable Complete M-partite Graphs

For a simple undirected graph G with vertex set V and edge set E, an m-total weighting λ of G is a mapping from V（G） ∪ E（G） to {1, 2, · · ·, m}. Every number in{1, 2, · · ·, m} is called the weight that is used in the m-total weighting of graph G.An m-total weighting of G is called an m-total irregular assignment if, for any two distinct vertices u and v, the sum of the weights of vertex u and the edges incident with u is different from the sum of the weights of vertex v and the edges incident with v. The minimum integer m such that G has an m-total irregular assignment is called the total vertex irregularity strength of G. The complete m-partite graph on n vertices in which each part has either n m or n m vertices is denoted by T_m,n. The total vertex irregularity strength of several equitable complete m-partite graphs, namely,T_m,m+1, T_m,m+2, T_m,m+3, T_m,2m, T_m,2m+1, T_m,3m-1（m ≥ 4） and T_m,n（n = 3m + r, r =1, 2, · · ·, m- 1）, are discussed in Chapter 2. We characterize completely the total vertex irregularity strength of equitable complete m-partite graphs T_3,n, T_4,n, T_5,n and T_6,n in Chapter 3.