Bounding the distant irregularity strength of graphs via a non-uniformly biased random weight assignment

Abstract

Given an edge $k$-weighting $\omega :E\to \left[k\right]$ of a graph $G=(V,E)$, the weighted degree of a vertex $v\in V$ is the sum of its incident weights. The least $k$ for which there exists an edge $k$-weighting such that the resulting weighted degrees of the vertices at distance at most $r$ in $G$ are distinct is called the $r$-distant irregularity strength, and denoted $s_r\left(G\right)$. This concept links the well-known 1–2–3 Conjecture, corresponding to $s_1\left(G\right)$, with the irregularity strength of graphs, $s\left(G\right)$, which coincides with $s_r\left(G\right)$ for every $r$ at least the diameter of $G$. It is believed that for every $r\ge 2$, $s_r\left(G\right)\le (1+o\left(1\right)){\Delta }^{r-1}$, where $\Delta$ is the maximum degree of $G$, while it is known that $s_r\left(G\right)\le 6{\Delta }^{r-1}$ in general and $s_r\left(G\right)\le (4+o\left(1\right)){\Delta }^{r-1}$ for graphs with minimum degree $\delta$ at least ${log}^8\Delta$. We apply the probabilistic method in order to improve these results and show that graphs with $\delta \gg ln\Delta$ satisfy $s_r\left(G\right)\le (e+o\left(1\right)){\Delta }^{r-1}$ as $\Delta \to \infty$.