Zhanping Yang | Search arXiv

arXiv2026

Intersections of Directed Graphs

Given two weighted directed graphs of order $n$, we study how much their overlap can deviate from its random average under relabelling, and how concentrated the distribution of their intersection is when they are placed at random. Bollobás and Scott studied these problems on weighted $k$-uniform hypergraphs and raised the corresponding questions for directed graphs. In this paper, we study these problems for weighted directed graphs and obtain results analogous to those in the weighted hypergraph setting.

math.CO↗

arXiv2026

Intersections of Oriented Graphs and Tournaments

Given two tournaments of order $n$, Bollobás and Scott defined their discrepancy as the largest deviation of their overlap from its random average under relabelling, and they asked whether the resulting discrepancy is always $Ω(n^{3/2})$. We answer this question by proving that there is an absolute constant $c>0$ such that every pair of tournaments $T,U$ of order $n$ has discrepancy at least $cn^{3/2}$. More generally, if $D$ and $H$ are oriented graphs of order $ n $ with $ e(D) = p \binom{n}{2} $ and $ e(H) = q \binom{n}{2} $ satisfying $16/n \leq p, q \leq 1 - 16/n$, then there is an absolute constant $ c > 0 $ such that their discrepancy is at least $c(p(1-p)q(1-q))^{3}n^{3/2}$. We also show that these two-graph estimates extend to intersections of any fixed number of graphs, tournaments, and oriented graphs.

math.CO↗