Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On two-coloring bipartite uniform hypergraphs

Boyoon Lee, Theodore Molla, Brendan Nagle

TL;DR

The paper addresses 2-coloring bipartite $k$-uniform hypergraphs by proving that, for fixed $k \ge 3$, one can construct a bipartition of any bipartite $k$-graph on $n$ vertices in average time $O(n^k)$ over the class ${\mathcal B}_n$. It provides two distinct proofs: an elementary probabilistic approach based on joint degrees and Chernoff concentration, and a constructive approach using the Szemer\'edi Regularity Lemma to obtain an approximate partition refined to a true bipartition. The work connects to prior results by Dyer-Frieze and Person-Schacht, and discusses implications for related Fano-plane design problems, including potential average-case improvements for the Fano-colored case. Together, these results yield practical average-case algorithms for bipartite 2-coloring in hypergraphs and offer insights that may extend to related design-theoretic structures.

Abstract

Of a given bipartite graph $G = (V, E)$, it is elementary to construct a bipartition in time $O(|V| + |E|)$. For a given $k$-graph $H = H^{(k)}$ with $k \geq 3$ fixed, Lovász proved that deciding whether $H$ is bipartite is NP-complete. Let $\mathcal{B}_n$ denote the collection of all $[n]$-vertex bipartite $k$-graphs. We construct, of a given $H \in \mathcal{B}_n$, a bipartition in time averaging $O(n^k)$ over the class $\mathcal{B}_n$. We provide two proofs of our result. When $k = 3$, this result expedites one of Person and Schacht.

On two-coloring bipartite uniform hypergraphs

TL;DR

The paper addresses 2-coloring bipartite -uniform hypergraphs by proving that, for fixed , one can construct a bipartition of any bipartite -graph on vertices in average time over the class . It provides two distinct proofs: an elementary probabilistic approach based on joint degrees and Chernoff concentration, and a constructive approach using the Szemer\'edi Regularity Lemma to obtain an approximate partition refined to a true bipartition. The work connects to prior results by Dyer-Frieze and Person-Schacht, and discusses implications for related Fano-plane design problems, including potential average-case improvements for the Fano-colored case. Together, these results yield practical average-case algorithms for bipartite 2-coloring in hypergraphs and offer insights that may extend to related design-theoretic structures.

Abstract

Of a given bipartite graph , it is elementary to construct a bipartition in time . For a given -graph with fixed, Lovász proved that deciding whether is bipartite is NP-complete. Let denote the collection of all -vertex bipartite -graphs. We construct, of a given , a bipartition in time averaging over the class . We provide two proofs of our result. When , this result expedites one of Person and Schacht.
Paper Structure (6 sections, 6 theorems, 77 equations)

This paper contains 6 sections, 6 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. An Elementary Proof of Theorem \ref{['thm:main']}
  3. A Second Algorithm for Theorem \ref{['thm:main']}
  4. A Second Proof of Theorem \ref{['thm:main']}
  5. Proof of Lemma \ref{['lem:firstsub']}
  6. Concluding Remarks

Key Result

Theorem A

There exists an algorithm $\mathbb{A}_{{\rm bip}}$ which constructs, for each fixed integer $k \geq 3$ and of a given $k$-graph $H \in {\mathcal{B}}_n$, a bipartition $[n] = X_H \, \dot{\cup} \, Y_H$ of $H$ in time averaging $O(n^k)$ over the class ${\mathcal{B}}_n$.

Theorems & Definitions (13)

  • Theorem A
  • Theorem 1.1: Person and Schacht PSalmPSalg
  • Corollary 1.2: Person and Schacht PSalmPSalg
  • Remark
  • Definition 2.1: $\boldsymbol{\sigma}$- standard
  • Remark 2.2
  • Theorem 3.1: Kohayakawa, Rödl, and Thoma KRT
  • Remark 3.2
  • Definition 4.1: $\boldsymbol{\delta}$-typicality
  • Remark 4.2
  • ...and 3 more