Graded discrepancy of graphs and hypergraphs

TL;DR

The paper addresses the graded discrepancy problem, seeking the smallest $c(p,n)$ such that there exists a vertex ordering with $|e(\{v_1,\dots,v_i\})-p\binom{i}{2}|\le c(p,n)$ for all $i$ in graphs (and a hypergraph generalization). It develops linear-in-$n$ upper bounds via two ordering-based algorithms and extends these results to $k$-uniform hypergraphs, establishing analogous bounds in terms of $\binom{n-1}{k-1}$. It also constructs extremal examples to derive lower bounds, demonstrating that the upper bounds are tight up to constant factors in many density regimes. Overall, the work advances the understanding of graded discrepancy in both graphs and hypergraphs, providing near-tight characterizations across different edge densities.

Abstract

This paper studies the following question of Bollobás and Scott: Let $G$ be a graph with $n$ vertices and $p\binom{n}{2}$ edges. What is the smallest $c(p, n)$ such that there is an ordering $v_1, \ldots, v_n$ of the vertices in $G$ with $\left|e(\{v_1, \ldots, v_i\})-p\binom{i}{2}\right|\leq c(p, n)$ for all $i\in \{1,\ldots,n\}$ ? We obtain upper and lower bounds for $c(p,n)$ that are both linear in $n$. Furthermore, we generalize the result to $k$-uniform hypergraphs.

Theorems & Definitions (6)

• Definition 1.2
• proof : Proof
• proof : Proof
• proof : Proof of Theorem \ref{['31']}
• proof : Proof of Theorem \ref{['33']}
• proof : Proof of Theorem \ref{['34']}