On sets of integers with restrictions on their products

TL;DR

This paper investigates the minimum range of integers needed for product-injective labelings of graphs with bounded degree, revealing asymptotic behaviors depending on the degree relative to the number of vertices.

Contribution

It determines the asymptotic values of P(n,d) for most degrees d, establishing clear thresholds for when the labeling range scales linearly or logarithmically with n.

Findings

P(n,d) is asymptotically n for small degrees d.

P(n,d) scales as n log n for larger degrees d.

The results cover almost all ranges of d relative to n.

Abstract

A {\em product-injective labeling} of a graph $G$ is an injection $\chi: V(G) \to \mathbb{Z}$ such that $\chi(u)\chi(v) \not= \chi(x)\chi(y)$ for any distinct edges $uv, xy\in E(G)$. Let $P(G)$ be the smallest $N \geq 1$ such that there exists a product-injective labeling $\chi : V(G) \rightarrow [N]$. Let $P(n,d)$ be the maximum possible value of $P(G)$ over $n$-vertex graphs $G$ of maximum degree at most $d$. In this paper, we determine the asymptotic value of $P(n,d)$ for all but a small range of values of $d$ relative to $n$. Specifically, we show that there exist constants $a,b > 0$ such that $P(n,d) \sim n$ if $d \leq \sqrt{n}(\log n)^{-a}$ and $P(n,d) \sim n\log n$ if $d \geq \sqrt{n}(\log n)^{b}$.