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Choosability of multipartite hypergraphs

Peter Bradshaw, Abhishek Dhawan, Nhi Dinh, Shlok Mulye, Rohan Rathi

Abstract

A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. We show that $k$-partite $k$-graphs of maximum degree $Δ$ are $q$-choosable for $q \geq \left(\frac{4}{5}(k-1 + o(1))Δ/\log Δ\right)^{1/(k-1)}$. Our proof yields an efficient randomized algorithm for finding such a coloring, which shows that the conjectured algorithmic barrier for coloring pseudorandom $k$-graphs does not apply to $k$-partite $k$-graphs.

Choosability of multipartite hypergraphs

Abstract

A -uniform hypergraph (or -graph) is -partite if can be partitioned into sets such that each edge in contains precisely one vertex from each . We show that -partite -graphs of maximum degree are -choosable for . Our proof yields an efficient randomized algorithm for finding such a coloring, which shows that the conjectured algorithmic barrier for coloring pseudorandom -graphs does not apply to -partite -graphs.
Paper Structure (9 sections, 10 theorems, 60 equations)

This paper contains 9 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background and results
  3. Relation to prior work
  4. Proof overview
  5. Preliminaries
  6. Proof of Theorem \ref{['theorem: constant inside']}
  7. Proofs of Propositions
  8. Probability bound
  9. Existence of $L^*$

Key Result

Theorem 1.3

The following holds for $\Delta$ sufficiently large. Let $G$ be a bipartite graph of maximum degree at most $\Delta$. Then,

Theorems & Definitions (22)

  • Definition 1.1
  • Conjecture 1.2: alon1998choice
  • Theorem 1.3: bradshaw2024bipartite
  • Theorem 1.4: dhawan2025list
  • Conjecture 1.5: dhawan2025list
  • Theorem 1.6
  • Theorem 2.1: Lovász Local Lemma; AlonSpencer
  • Theorem 2.2: Harris's Inequality/Kleitman's Lemma AlonSpencer
  • Theorem 2.3: Jensen's Inequality hardy1952inequalities
  • Proposition 3.1
  • ...and 12 more