Graded discrepancy of graphs and hypergraphs

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.