Betti numbers of order-preserving graph homomorphisms

Abstract

For graphs $G$ and $H$ with totally ordered vertex sets, a function mapping the vertex set of $G$ to the vertex set of $H$ is an order-preserving homomorphism from $G$ to $H$ if it is nondecreasing on the vertex set of $G$ and maps edges of $G$ to edges of $H$. In this paper, we study order-preserving homomorphisms whose target graph $H$ is the complete graph on $n$ vertices. By studying a family of graphs called nonnesting arc diagrams, we are able to count the number of order-preserving homomorphisms (and more generally the number of order-preserving multihomomorphisms) mapping any fixed graph $G$ to the complete graph $K_n$.