Vertex partition of hypergraphs and maximum degenerate subhypergraphs

Abstract

In 2007 Matamala proved that if $G$ is a simple graph with maximum degree $\Delta \ge 3$ not containing $K_{\Delta +1}$ as a subgraph and $s,t$ are positive integers such that $s+t\ge \Delta$, 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 Bollobás and Manvel, by Catlin, by Gerencsér 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.

12 pages