## Involve: A Journal of Mathematics

- Involve
- Volume 5, Number 1 (2012), 67-80.

### 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 ${K}_{n}$.

#### Article information

**Source**

Involve, Volume 5, Number 1 (2012), 67-80.

**Dates**

Received: 27 May 2011

Accepted: 11 July 2011

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513733450

**Digital Object Identifier**

doi:10.2140/involve.2012.5.67

**Mathematical Reviews number (MathSciNet)**

MR2924315

**Zentralblatt MATH identifier**

1246.13017

**Subjects**

Primary: 13D02: Syzygies, resolutions, complexes

Secondary: 05A18: Partitions of sets 06A06: Partial order, general 05C30: Enumeration in graph theory

**Keywords**

graph homomorphisms Betti numbers nonnesting partitions

#### Citation

Guerra, Lauren; Klee, Steven. Betti numbers of order-preserving graph homomorphisms. Involve 5 (2012), no. 1, 67--80. doi:10.2140/involve.2012.5.67. https://projecteuclid.org/euclid.involve/1513733450