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  • Author: Thomas Tu x
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Abstract

A smooth hypergraph property Ρ is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs. In this paper we examine P-colorings of hypergraphs with smooth hypergraph properties P. A P-coloring of a hypergraph H with color set C is a function φ: V (H) → C such that H[φ−1(c)] belongs to Ρ for all c ∈ C. Let L : V (H) → 2C be a so called list-assignment of the hypergraph H. Then, a (P,L)-coloring of H is a P-coloring φ of H such that φ(v) ∈ L(v) for all v ∈ V (H). The aim of this paper is a characterization of (P,L)-critical hypergraphs. Those are hypergraphs H such that H − v is (P,L)-colorable for all v ∈ V (H) but H itself is not. Our main theorem is a Gallai-type result for critical hypergraphs, which implies a Brooks-type result for (P,L)-colorable hypergraphs. In the last section, we prove a Gallai-type bound for the degree sum of (P,L)-critical locally simple hypergraphs.

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