Total Vertex Irregularity Strength of Certain Equitable Complete $m$-Partite Graphs

Abstract

For a simple undirected graph $G$ with vertex set $V$ and edge set $E$, a total $k$-labeling $\lambda :V\cup E\to \{1,2,\dots ,k\}$ is called a vertex irregular total $k$-labeling of $G$ if for every two distinct vertices $x$ and $y$ of $G$, their weights $wt(x)$ and $wt(y)$ are distinct, where the weight of a vertex $x$ in $G$ is the sum of the label of $x$ and the labels of all edges incident with the vertex $x$. The total vertex irregularity strength of $G$, denoted by $\text{tus}(G)$, is the minimum $k$ for which the graph $G$ has a vertex irregular total $k$-labeling. The complete $m$-partite graph on $n$ vertices in which each part has either $\lfloor \frac{n}{m}\rfloor$ or $\lceil \frac{n}{m}\rceil$ vertices is denoted by $T_{n,m}$. The total vertex irregularity strength of some equitable complete $m$-partite graphs, namely, $T_{m,m+1}$, $T_{m,m+2}$, $T_{m,2m}$, $T_{m,2m+4}$, $T_{3m-1}$ ($m\ge 4$), $T_n$ ($n=3m+r$, $r=1,2,\dots ,m-1$), and equitable complete $3$-partite graphs have been studied in this paper.