Distant total irregularity strength of graphs via random vertex ordering

Let $c:V\cup E\to \{1,2,\dots ,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours, i.e. $c\left(u\right)+{\sum }_{e?u}c\left(e\right)\ne c\left(v\right)+{\sum }_{e?v}c\left(e\right)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at distance $1\le d(u,v)\le r$ in $G$ are sum distinguished is denoted by ${\mathrm{ts}}_r\left(G\right)$. Such graph invariants link the concept of the total vertex irregularity strength of graphs with so-called 1-2-Conjecture, whose concern is the case of $r=1$. Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that ${\mathrm{ts}}_r\left(G\right)\le (2+o\left(1\right)){\Delta }^{r?1}$ for every integer $r\ge 2$ and each graph $G$, thus improving the previously best result: ${\mathrm{ts}}_r\left(G\right)\le 3{\Delta }^{r?1}$.