Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

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, α({G}), (ii) acyclic orientations with a single source vertex, α 0 ({G}), and (iii) totally cyclic orientations, β({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 α(tri), α 0 (tri), and β(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 α({G}), α 0 ({G}), and β({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 τ ({G}) for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.

Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices

“…In this section we explain a method that we use to infer lower and upper bounds on the exponential growth constants φ(Λ) and σ(Λ). Our method is the same as the one that we have used in previous work [5][6][7] to infer lower and upper bounds on exponential growth constants for other graph-theoretic quantities, such as acyclic orientations, α(Λ), acyclic orientations with a unique source vertex, α 0 (Λ), and totally cyclic orientations, β(Λ), of directed graphs, and so we refer the reader to these previous works for further details.…”

“…This is the procedure that we used in Refs. [6,7] for certain exponential growth constants describing acyclic and cyclic orientations of edge arrows on directed lattice graphs. We list these approximate values in Table XXX.…”

“…This was also the case with our earlier calculations of bounds on exponential growth constants for acyclic orientations, acyclic orientations with a unique source, and totally cyclic orientations for directed lattices in Refs. [6,7]. This paper is organized as follows.…”

Exponential Growth Constants for Spanning Forests on Archimedean Lattices: Values and Comparisons of Upper Bounds

“…As we noted, our upper bounds on φ(Λ) for this and the other Archimedean lattices that we studied are, to our knowledge, the best upper bounds on φ(Λ) for these lattices. Our results in [1] were part of a general program of calculating bound on, and values of, exponential growth constants for various classes of subgraphs on Archimedean lattices [8][9][10].…”

“…In earlier work preceding [1,9,10], we had calculated values of exponential growth constants for a variety of families of lattice strip graphs of various lattices with a range of finite widths and with arbitrarily great length (e.g., [14]- [19]). We found that for a given type of lattice strip graph, in the infinite-length limit, φ is a monotonically increasing function of the strip width.…”