Total Vertex Irregular Labeling of Complete Bipartite Graphs

Abstract

A total vertex irregular labeling of a graph $G$ with $v$ vertices and $e$ edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum value of the largest label over all such irregular assignments. In this paper, we consider the total vertex irregular labeling of complete bipartite graphs $K_{m,n}$ and prove that

$$tvs(K_{m,n})\ge max\{\lceil \frac{m+n}{m+1}\rceil ,\lceil \frac{2m+n-1}{n}\rceil \}\text{if }(m,n)\ne (2,2).$$