Partitions of hypergraphs under variable degeneracy constraints

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 hy

Vertex partition of hypergraphs and maximum degenerate subhypergraphs

“…Thus, it is sufficient to consider only connected hypergraphs. The next result was proved by Schweser and Stiebitz [19]; for the class of simple graphs it was proved in 2000 by Borodin, Kostochka and Toft [4]. Theorem 1.1.…”

“…On the one hand, Theorem 1.1 is a strengthening of the result by Borodin, respectively Bollobás and Manvel. On the other hand, as explained in [19], Theorem 1.1 implies several well known result about colorings and list-colorings of graphs, respectively hypergraphs; in particular, the characterization of degree choosable graphs obtained by Erdős, Rubin, and Taylor [9] and the characterization of degree choosable hypergraphs given by Kostochka, Stiebitz, and Wirth [12]. The special case when p = ∆(G) and f i (v) = 1 for all v ∈ V (G) and 1 ≤ i ≤ p yields a Brooks-type result for hypergraphs which was obtained by Jones [11].…”

“…Graphs and hypergraphs considered in this paper may have parallel edges, but no loops. We will mainly use the notation from the paper [19]. Let G be a hypergraph.…”

Point partition numbers: decomposable and indecomposable critical graphs

“…The papers by Kronk and Mitchen [17], by Bollobás and Harary [1], by Mihok [20], and by Škrekovski [25] are all devoted to the structure of χ 2 -critical simple graphs. The papers by Schweser [22] and by Schweser and Stiebitz [23] contain some results about χ t -critical graphs having multiple edges and with arbitrary t ≥ 1.…”

“…The vertices of G having degree t(k − 1) are called low vertices of G, and the remaining vertices are called high vertices of G. So any high vertex of G has degree at least t(k − 1) + 1 in G. Furthermore, the subgraph of G induced by its low vertices is called the low vertex subgraph of G. For χ-critical graphs, this classification is due to Gallai [10]. The following result due to Schweser [22,Theorem 3] (see also Schweser and Stiebitz [23]) generalizes Gallai's theorem about the structure of the low vertex subgraph of χ-critical graphs. Theorem 3.6 (Schweser 2019).…”