Total Vertex Irregular Labeling of Complete Bipartite Graphs

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

\[
tvs(K_{m,n}) \geq \max \left\{ \left\lceil \frac{m+n}{m+1} \right\rceil, \left\lceil \frac{2m+n-1}{n} \right\rceil \right\} \quad \text{if } (m,n) \neq (2,2).
\]