Vertex partition of hypergraphs and maximum degenerate subhypergraphs

TL;DR

This paper extends a graph vertex partition theorem to hypergraphs, allowing for variable degeneracy and multiple parts, generalizing previous results and providing new insights into hypergraph structure.

Contribution

It introduces a hypergraph version of a vertex partition theorem, extending it to variable degeneracy and multiple partitions, advancing hypergraph theory.

Findings

Proved a hypergraph analogue of Matamala's theorem.

Extended the theorem to variable degeneracy.

Generalized the result to partitions into more than two parts.

Abstract

In 2007 Matamala proved that if $G$ is a simple graph with maximum degree $\Delta\geq 3$ not containing $K_{\Delta +1}$ as a subgraph and $s, t$ are positive integers such that $s+t \geq \Delta$, then the vertex set of $G$ admits a partition $(S,T)$ such that $G[S]$ is a maximum order $(s-1)$-degenerate subgraph of $G$ and $G[T]$ is a $(t-1)$-degenerate subgraph of $G$. This result extended earlier results obtained by Borodin, by Bollob\'as and Manvel, by Catlin, by Gerencs\'{e}r and by Catlin and Lai. In this paper we prove a hypergraph version of this result and extend it to variable degeneracy and to partitions into more than two parts, thereby extending a result by Borodin, Kostochka, and Toft.