Subgraph discrepancies in the complete graph

TL;DR

The paper addresses discrepancy in 2-colorings of $K_n$ for general guest graphs $F$, extending the EFLS framework from trees to graphs with $\Delta(F)\le (1-\varepsilon)n$ and no isolates, proving a linear discrepancy bound $c\,\varepsilon n$ and, for $d$-regular $F$, a bound $c\sqrt{\varepsilon d}\,n$ that is tight. It then determines optimal constants for $K_k$-factors via a bipartite construction (defining $\lambda_k$) and for $2$-factors, achieving $(2/3-o(1))n$ monochromatic edges with tight extremal examples. The core technique combines a biased cuts analysis with a main embedding lemma within the guest-good/host-good framework, enabling probabilistic embedding and careful switching arguments to force large color imbalances in copies of $F$. The results advance understanding of how graph structure (e.g., max degree, regularity, factor structure) governs unavoidable discrepancy, and they suggest precise extremal configurations (notably bipartite constructions) that achieve or limit these bounds. The work also opens questions on stability and extensions to more colors or broader graph families.

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 $\left| \sum_{e \in E(F)} f(e) \right|$. Erdős, Füredi, Loebl and Sós 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 $Ω(\varepsilon)n$. 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 $Ω(\sqrt{\varepsilon d}) \cdot n$. 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 $λ$ such that every 2-coloring of $K_n$ has a copy of $F$ with discrepancy at least $(λ+ o(1))n$.

Figures (1)

• Figure 1: First few values of $\rho_k$ and $\lambda_k$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: bipartite construction
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Theorem 2.4: LLS
• Theorem 2.5: Follows from Bollobas_ScottRST
• Definition 2.6: guest-good