On the number of planar Eulerian orientations

Abstract

The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.

The generating function of planar Eulerian orientations

“…Right: the same map, equipped with an Eulerian orientation. a way that every vextex has as many incoming as outgoing edges, see Figure 1) was raised by Bonichon et al [13]. They did not solve the problem, but gave sequences of lower bounds and upper bounds on the number of planar Eulerian orientations.…”

Counting planar Eulerian orientations

“…The latter is in the universality class of the celebrated sixvertex model on a random lattice, a problem that has been studied by Kostov [8] and Zinn-Justin [10]. Unfortunately, the nature of their solutions are not in the form of a generating function that can be compared to the enumerative results of 1 email: andrewelveyprice@gmail.com 2 email: guttmann@unimelb.edu.au 1 These are planar maps in which the degree of every vertex is even.…”

“…Recently the problem of enumerating planar Eulerian orientations with n edges was considered by Bonichon, Bousquet-Mélou, Dorbec and Pennarun [1]. Enumeration of Eulerian orientations on a given graph has been previously considered, for example by Felsner and Zickfeld [3] who established rigorous bounds on the growth constant for these and other combinatorial structures.…”

“…for Eulerian orientations counted by edges and µ = 4 √ 3π for 4-valent Eulerian orientations counted by vertices. In [1], a different approach was taken. Families of subsets and supersets were counted.…”

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Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun [1], we give a system of functional equations which characterise the ordinary generating function, U (x), for the number of planar Eulerian orientations counted by edges. We also characterise the ogf A(x), for 4-valent planar Eulerian orientations counted by vertices in a similar way.

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