Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

TL;DR

This paper establishes tight bounds on exponential growth constants related to orientations of heteropolygonal Archimedean lattices, revealing their monotonic increase with lattice coordination and supporting existing conjectures.

Contribution

It provides the first tight bounds on these growth constants for heteropolygonal Archimedean lattices and demonstrates their monotonic relationship with lattice coordination.

Findings

Bounds are close, enabling accurate estimation of growth constants.

Growth constants increase monotonically with lattice coordination.

Results support the Merino-Welsh and Conde-Merino conjectures.

Abstract

We infer upper and lower bounds on the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several $n$-vertex heteropolygonal Archimedean lattices $\Lambda$. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.