Partitions of hypergraphs under variable degeneracy constraints

TL;DR

This paper establishes conditions for partitioning hypergraphs into subhypergraphs with variable degeneracy constraints, introducing the concept of hard pairs which characterize when such partitions are possible.

Contribution

The paper introduces the concept of hard pairs, providing a complete characterization for the existence of hypergraph partitions under variable degeneracy constraints.

Findings

Partitions exist iff the hypergraph is not a hard pair.

Hard pairs are recursively defined configurations.

Applications to generalized hypergraph coloring problems.

Abstract

The paper deals with partitions of hypergraphs into induced subhypergraphs satisfying constraints on their degeneracy. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph $H$ and a sequence $f=(f_1,f_2, \ldots, f_p)$ of $p\geq 1$ vertex functions $f_i:V(H) \to \mathbb{N}_0$ such that $f_1(v)+f_2(v)+ \cdots + f_p(v)\geq d_H(v)$ for all $v\in V(H)$, we want to find a sequence $(H_1,H_2, \ldots, H_p)$ of vertex disjoint induced subhypergraphs containing all vertices of $H$ such that each hypergraph $H_i$ is strictly $f_i$-degenerate, that is, for every non-empty subhypergraph $H'\subseteq H_i$ there is a vertex $v\in V(H')$ such that $d_{H'}(v)<f_i(v)$. Our main result in this paper says that such a sequence of hypergraphs exists if and only if $(H,f)$ is not a so-called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. Our main result has several interesting applications related to generalized hypergraph coloring problems.