Abstract
Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.
Highlights
Definition: An r-uniform hypergraph G consists of the vertex set V(G) and the edge set E(G) which is a collection of r-subsets of Definition: An r-uniform hypergraph G consists of the vertex set V(G) and the edge set E(G) which is a collection of r-subsets of
Example: A 2-coloring of a 3-uniform hypergraph which is not realized by any homomorphism
Hom complexes never work for colorings if we work in the category of r-uniform hypergraphs and homomorphisms between them
Summary
Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of. A homomorphism f:G→H is a map f:V(G)→V(H) whenever {v1,...,vr}∈E(G),. Get a category of r-uniform hypergraphs and homomorphisms
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