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On high discrepancy $1$-factorizations of complete graphs

Jiangdong Ai, Fankang He, Seonghyuk Im, Hyunwoo Lee

TL;DR

The paper proves that for all sufficiently large $n$, every edge-signed complete graph $K_{2n}$ admits a $1$-factor decomposition in which each perfect matching has a uniformly positive discrepancy, i.e., disc$(\psi_i) \ge c$ for a universal $c>0$. The approach combines randomization with concentration inequalities (notably Talagrand-type bounds for random permutations) and a switcher mechanism based on a decomposition of $K_{2n}$ into small factors (e.g., $C_4$- and $K_4$-factors). It also provides a stronger balanced-coloring variant that, with high probability, aligns the discrepancies of all factors to within $\varepsilon$ of the global discrepancy, and it extends to unbalanced and multi-color generalizations. The results advance discrepancy theory for graph decompositions and introduce techniques potentially applicable to Hamilton-cycle decompositions and related Dirac-type questions in edge-colored graphs.

Abstract

We proved that for every sufficiently large $n$, the complete graph $K_{2n}$ with an arbitrary edge signing $σ: E(K_{2n}) \to \{-1, +1\}$ admits a high discrepancy $1$-factor decomposition. That is, there exists a universal constant $c > 0$ such that every edge-signed $K_{2n}$ has a perfect matching decomposition $\{ψ_1, \ldots, ψ_{2n-1}\}$, where for each perfect matching $ψ_i$, the discrepancy $\lvert \frac{1}{n} \sum_{e\in E(ψ_i)} σ(e) \rvert$ is at least $c$.

On high discrepancy $1$-factorizations of complete graphs

TL;DR

The paper proves that for all sufficiently large , every edge-signed complete graph admits a -factor decomposition in which each perfect matching has a uniformly positive discrepancy, i.e., disc for a universal . The approach combines randomization with concentration inequalities (notably Talagrand-type bounds for random permutations) and a switcher mechanism based on a decomposition of into small factors (e.g., - and -factors). It also provides a stronger balanced-coloring variant that, with high probability, aligns the discrepancies of all factors to within of the global discrepancy, and it extends to unbalanced and multi-color generalizations. The results advance discrepancy theory for graph decompositions and introduce techniques potentially applicable to Hamilton-cycle decompositions and related Dirac-type questions in edge-colored graphs.

Abstract

We proved that for every sufficiently large , the complete graph with an arbitrary edge signing admits a high discrepancy -factor decomposition. That is, there exists a universal constant such that every edge-signed has a perfect matching decomposition , where for each perfect matching , the discrepancy is at least .

Paper Structure

This paper contains 9 sections, 10 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Concentration inequalities
  5. Decompositions of complete graphs
  6. Proof of main results
  7. Proof of Theorem \ref{['thm:balanced_coloring']}
  8. Proof of Theorem \ref{['thm:high_discrepancy']}
  9. Concluding remark

Key Result

Theorem 1.2

There exists a universal constant $c>0$ such that the following holds for every sufficiently large $n$. For every edge coloring $\sigma:E(K_{2n}) \to \{-1, +1\}$ of the complete graph on $2n$ vertices has a $1$-factor decomposition $\{\psi_1, \ldots, \psi_{2n-1}\}$ of $K_{2n}$ such that for each $i

Figures (1)

  • Figure 1: Six kinds of colored $C_4$

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Chebyshev's inequality
  • Lemma 2.2: Talagrand's inequality on random permutations McDiarmid-Talagrand
  • Lemma 2.3
  • Lemma 3.1
  • proof : Proof of \ref{['lem:balanced_coloring']}
  • Claim 3.2
  • Theorem 3.3
  • ...and 10 more