Strongly polynomial sequences as interpretations

TL;DR

This paper introduces a new model-theoretic approach using interpretation schemes to construct strongly polynomial graph sequences, unifying previous methods and suggesting all such sequences may be generated by this framework.

Contribution

It presents a general interpretation-based method for constructing strongly polynomial graph sequences, extending and unifying prior constructions in the field.

Findings

The interpretation scheme method encompasses all known constructions.

The conjecture that all strongly polynomial sequences can be generated by this method.

Verification of the conjecture for sequences with bounded degree.

Abstract

A strongly polynomial sequence of graphs $(G_n)$ is a sequence $(G_n)_{n\in\mathbb{N}}$ of finite graphs such that, for every graph $F$, the number of homomorphisms from $F$ to $G_n$ is a fixed polynomial function of $n$ (depending on $F$). For example, $(K_n)$ is strongly polynomial since the number of homomorphisms from $F$ to $K_n$ is the chromatic polynomial of $F$ evaluated at $n$. In earlier work of de la Harpe and Jaeger, and more recently of Averbouch, Garijo, Godlin, Goodall, Makowsky, Ne\v{s}et\v{r}il, Tittmann, Zilber and others, various examples of strongly polynomial sequences and constructions for families of such sequences have been found. We give a new model-theoretic method of constructing strongly polynomial sequences of graphs that uses interpretation schemes of graphs in more general relational structures. This surprisingly easy yet general method encompasses all previous constructions and produces many more. We conjecture that, under mild assumptions, all strongly polynomial sequences of graphs can be produced by the general method of quantifier-free interpretation of graphs in certain basic relational structures (essentially disjoint unions of transitive tournaments with added unary relations). We verify this conjecture for strongly polynomial sequences of graphs with uniformly bounded degree.