# Distant total irregularity strength of graphs via random vertex ordering

## TL;DR

This paper establishes an improved probabilistic bound on the total irregularity strength of graphs for vertices within a certain distance, advancing understanding of graph colorings related to the 1-2-Conjecture.

## Contribution

It combines probabilistic and combinatorial methods to prove a tighter upper bound on the total irregularity strength for any graph and distance parameter.

## Key findings

- Proves that ${
m ts}_r(G) o (2+o(1))\Delta^{r-1}$ for all graphs and r ≥ 2.
- Improves previous bound of ${
m ts}_r(G) o 3\Delta^{r-1}$.
- Uses a novel combination of probabilistic and combinatorial techniques.

## Abstract

Let $c:V\cup E\to\{1,2,\ldots,k\}$ 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. $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at distance $1\leq d(u,v)\leq r$ in $G$ are sum distinguished is denoted by ${\rm ts}_r(G)$. 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 ${\rm ts}_r(G)\leq (2+o(1))\Delta^{r-1}$ for every integer $r\geq 2$ and each graph $G$, thus improving the previously best result: ${\rm ts}_r(G)\leq 3\Delta^{r-1}$.