Choosability of multipartite hypergraphs

TL;DR

This paper proves that $k$-partite $k$-uniform hypergraphs with bounded degree are $q$-choosable for a specific $q$, providing an efficient randomized coloring algorithm that challenges previous conjectures about pseudorandom hypergraph coloring.

Contribution

It establishes new bounds on list coloring for $k$-partite $k$-graphs and introduces an efficient algorithm, contradicting prior beliefs about coloring barriers.

Findings

$k$-partite $k$-graphs are $q$-choosable under certain degree conditions

An efficient randomized coloring algorithm is developed

Challenges existing conjectures on pseudorandom hypergraph coloring

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 $\Delta$ are $q$-choosable for $q \geq \left(\frac{4}{5}(k-1 + o(1))\Delta/\log \Delta\right)^{1/(k-1)}$. 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.