Authors: Kevin Buchin ¹; Bettina Speckmann ¹; Sander Verdonschot ²

• 1 Department of mathematics and computing science [Eindhoven]
• 2 School of Computer Science (Carleton, Ottawa)

We prove that any irreducible triangulation on n vertices has O(4.6807n) regular edge labelings and that there are irreducible triangulations on n vertices with Ω(3.0426n) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O(1.87n).