Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs

TL;DR

This paper calculates and bounds exponential growth constants for orientations of directed lattice graphs, providing new precise estimates and exact values, and explores their asymptotic behavior and relation to spanning trees.

Contribution

It introduces the best current bounds and exact values for exponential growth constants of acyclic and cyclic orientations on various lattice graphs, advancing understanding of their asymptotic properties.

Findings

Bounds on exponential growth constants are very close to exact values.

Exponential growth constants increase monotonically with vertex degree.

Ratios of orientation quantities to total edge orientations approach asymptotic limits.

Abstract

We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs $G$ of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, $\alpha(\{G\})$, (ii) acyclic orientations with a single source vertex, $\alpha_0(\{G\})$, and (iii) totally cyclic orientations, $\beta(\{G\})$. We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from $O(10^{-4})$ to $O(10^{-2})$. Further, we present exact values of $\alpha(tri)$, $\alpha_0(tri)$, and $\beta(hc)$ and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted $sq_d$. We show that $\alpha(\{G\})$, $\alpha_0(\{G\})$, and $\beta(\{G\})$ are monotonically increasing functions of vertex degree for these lattices. We also study the asymptotic behavior of the ratios of the quantities (i)-(iii) divided by the total number of edge orientations as the number of vertices goes to infinity. A comparison is given of these exponential growth constants with the corresponding exponential growth constant $\tau(\{G\})$ for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.