Sum-distinguishing number of sparse hypergraphs

TL;DR

This paper investigates the sum-distinguishing number of sparse hypergraphs, establishing near-quadratic bounds in relation to the number of hyperedges, and explores implications for hypergraph irregularity strength and neighborhood distinguishing parameters.

Contribution

It provides the first near-quadratic bounds on s(n,m) for sparse hypergraphs and connects these bounds to hypergraph irregularity strength and neighborhood distinguishing numbers.

Findings

s(n,m) is approximately m^2/w(m) for certain ranges of m

Hypergraphs with irregularity strength exceeding 2n exist, reaching at least n^{2-o(1)}

The neighborhood distinguishing number s*(n) is also near-quadratic in n

Abstract

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n < m < n^{O(1)} then s(n,m)= m^2/w(m), where w(m) is a function that goes to infinity and is smaller than any polynomial in m. The parameter s(n,m) has close connections to several other graph and hypergraph functions, such as the irregularity strength of hypergraphs. Our result has several applications, notably: 1. We answer a question of Gyarfas et al. whether there are n-vertex hypergraphs with irregularity strength greater than 2n. In fact we show that there are n-vertex hypergraphs with irregularity strength at least n^{2-o(1)}. 2. Our results imply that s*(n)=n^2/w(n) where s*(n) is the distinguishing closed-neighborhood number, i.e., the smallest integer N such that any n-vertex graph allows for a vertex labeling with positive integers at most N so that the sums of labels on distinct closed neighborhoods of vertices are distinct.