Elsevier

Discrete Mathematics

Volume 341, Issue 9, September 2018, Pages 2567-2574
Discrete Mathematics

Homomorphism complexes and k-cores

https://doi.org/10.1016/j.disc.2018.06.014Get rights and content
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Abstract

For any fixed graph G, we prove that the topological connectivity of the graph homomorphism complex Hom(G,Km) is at least mD(G)2, where D(G)=maxHGδ(H), for δ(H) the minimum degree of a vertex in a subgraph H. This generalizes a theorem of C̆ukić and Kozlov, in which the maximum degree Δ(G) was used in place of D(G), and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, χ(G)D(G)+1, as χ(G)=min{m:Hom(G,Km)}. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom(G(n,p),Km) when p=cn for a fixed constant c>0.

Keywords

Hom-complexes
k-cores
Degeneracy
Random polyhedral complexes

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1

Present address: Department of Mathematics, Duke University, Durham, NC 27708, USA.