## On the total vertex-irregular strength of a disjoint union of $$t$$ copies of a path.(English)Zbl 1197.05130

Summary: For a simple graph $$G$$ with the vertex set $$V$$ and the edge set $$E$$, a labelling $$\lambda:V\cup E\to\{1,2,3,\dots,k\}$$ is called a vertex-irregular total $$k$$-labelling of $$G$$ if for any two different vertices $$x$$ and $$y$$ in $$V$$ we have $$wt(x)\neq wt(y)$$ where $$wt(x)= \lambda(x)+ \sum_{xy\in E}\lambda(xy)$$. The total vertex-irregular strength, denoted by $$tvs(G)$$, is the small est positive integer $$k$$ for which $$G$$ has a vertex-irregular total $$k$$-labelling. In this paper, we determine the total vertex-irregular strength of a disjoint union of t copies of a path, denoted by $$tP_n$$. We prove that for any $$t\geq 2$$,
$tvs(tP_n)=\begin{cases} t &\text{for }n=1,\\ t+1 &\text{for }2\leq n\leq3,\\ \big\lceil\frac{nt+1}{3}\big\rceil &\text{for }n\geq 4. \end{cases}$

### MSC:

 05C78 Graph labelling (graceful graphs, bandwidth, etc.)

### Keywords:

path; total $$k$$-labeling; total vertex-irregular strength