Subgraph discrepancies in the complete graph

TL;DR

This paper extends previous results on subgraph discrepancy in 2-edge-colored complete graphs, establishing bounds for various graph classes including general graphs with bounded degree, regular graphs, and specific structures like factors.

Contribution

It generalizes known discrepancy results from trees to broader classes of graphs, including those with maximum degree constraints and regular graphs, and determines optimal constants for specific graph factors.

Findings

Discrepancy bounds for graphs with maximum degree at most (1-ε)n

Discrepancy bounds for d-regular graphs with d ≤ (1-ε)n

Optimal discrepancy constants for K_r-factors and 2-factors

Abstract

Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erd\H{o}s, F\"uredi, Loebl and S\'os showed that if $F$ is an $n$-vertex tree with maximum degree at most $(1-\varepsilon)n$, then every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $\Omega(\varepsilon)n$. We extend this result by showing that the same conclusion holds for every $n$-vertex graph with maximum degree at most $(1-\varepsilon)n$ and no isolated vertices. We also show that for every $d$-regular $n$-vertex graph $F$ with $d \leq (1-\varepsilon)n$, every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $\Omega(\sqrt{\varepsilon d}) \cdot n$. The dependence on $d$ and $n$ is best possible. Finally, we consider specific graphs $F$, namely $K_r$-factors and 2-factors. For each such graph $F$, we determine the optimal constant $\lambda$ such that every 2-coloring of $K_n$ has a copy of $F$ with discrepancy at least $(\lambda + o(1))n$.