Subgraph discrepancies in the complete graph

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 $(\lambda + o(1))n$.