Graded discrepancy of graphs and hypergraphs

TL;DR

This paper investigates the minimal discrepancy in vertex orderings of graphs and hypergraphs, providing bounds that are linear in the number of vertices, thus advancing understanding of graph and hypergraph structure.

Contribution

It establishes linear bounds for the discrepancy measure in graphs and extends these results to hypergraphs, addressing a question posed by Bollobás and Scott.

Findings

Derived upper and lower bounds for discrepancy in graphs

Extended discrepancy bounds to hypergraphs

Bounds are both linear in the number of vertices

Abstract

This paper studies the following question of Bollob\'as 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.