Criteria for the less-equal-relation between partial Lov\'asz-vectors of digraphs

TL;DR

This paper establishes criteria and methods for comparing the number of homomorphisms between digraphs, linking partial Lovász-vectors to homomorphism counts, with applications to posets and undirected graphs.

Contribution

It introduces criteria for less-equal relations between partial Lovász-vectors of digraphs and develops a method for rearranging digraphs to compare homomorphism counts.

Findings

Relations between homomorphism counts are characterized for various classes of digraphs.

A method for rearranging digraphs to compare homomorphism counts is developed.

Results extend to undirected graphs and posets.

Abstract

Finite digraphs $R$ and $S$ are studied with $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every finite digraph $G \in \mathfrak{ D }'$, where ${\cal H}(G,H)$ is the set of order homomorphisms from $G$ to $H$ and $\mathfrak{ D }'$ is a class of finite digraphs. It is shown that for several classes $\mathfrak{ D }'$ of digraphs and $R \in \mathfrak{ D }'$, the relation $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every $G \in \mathfrak{ D }'$ is implied by the relation $\# {\cal S}(G,R) \leq \# {\cal S}(G,S)$ for every $G \in \mathfrak{ D }'$, where ${\cal S}(G,H)$ is the set of homomorphisms from $G$ to $H$ mapping all proper arcs of $G$ to proper arcs of $H$. Under an application-oriented regularity condition, the two relations are even equivalent. A method is developed for the rearrangement of a digraph $R$, resulting in a digraph $S$ with $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every digraph $G$. The method is applied in constructing pairs of partially ordered sets $R$ and $S$ with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for every partially ordered set $P$. The main part of the results holds also for undirected graphs.