# Distant irregularity strength of graphs with bounded minimum degree

## TL;DR

This paper establishes an improved upper bound on the $r$-distant irregularity strength of graphs with bounded minimum degree using probabilistic methods, linking it to the 1--2--3 Conjecture.

## Contribution

The paper introduces a tighter upper bound for $s_r(G)$ for graphs with minimum degree at least $ ext{ln}^8 riangle$, advancing previous bounds significantly.

## Key findings

- Proves $s_r(G) extless= (4+o(1)) riangle^{r-1}$ for graphs with minimum degree $ ext{ln}^8 riangle$.
- Uses probabilistic method to improve bounds on graph irregularity strength.
- Connects the $r$-distant irregularity strength to the 1--2--3 Conjecture.

## Abstract

Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any integer $r\geq 2$, the least $k$ admitting the existence of such $c$ attributing distinct weighted degrees to any two different vertices at distance at most $r$ in $G$ is called the $r$-distant irregularity strength of $G$ and denoted by $s_r(G)$. This graph invariant provides a natural link between the well known 1--2--3 Conjecture and irregularity strength of graphs. In this paper we apply the probabilistic method in order to prove an upper bound $s_r(G)\leq (4+o(1))\Delta^{r-1}$ for graphs with minimum degree $\delta\geq \ln^8\Delta$, improving thus far best upper bound $s_r(G)\leq 6\Delta^{r-1}$.