Eulerian orientations and the six-vertex model on planar map

TL;DR

This paper explores the enumeration of planar 4-valent maps with Eulerian orientations using two methods, connecting combinatorial enumeration with the six-vertex model from statistical physics.

Contribution

It introduces two distinct enumeration methods for Eulerian orientations on planar maps and establishes a connection between them and the six-vertex model.

Findings

Enumeration formulas for Eulerian orientations derived

Bijection between restricted and general Eulerian orientations shown

Generalization of enumeration results to include vertex weights

Abstract

We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.