On the number of regular edge labelings

Kevin Buchin,Bettina Speckmann,Sander Verdonschot

doi:10.46298/dmtcs.2085

Abstract

Combinatorics 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).

Highlights

An irreducible triangulation is a plane graph G such that (i) G is triangulated and the exterior face is a quadrangle, and (ii) G has no separating triangles (a 3-cycle with vertices both inside and outside the cycle)
A regular edge labeling of an irreducible triangulation G is a partition of the interior edges of G into two subsets of red and blue directed edges such that: (i) around each inner vertex in clockwise order we have one contiguous non-empty set each of incoming blue edges, outgoing red edges, outgoing blue edges, and incoming red edges; (ii) the left exterior vertex has only outgoing blue edges, the top exterior vertex has only incoming red edges, the right exterior vertex has only incoming blue edges, and the bottom exterior vertex has only outgoing red edges
A rectangular partition or floorplan is a partition of a rectangle into rectangular faces such that no four rectangles meet at a com- Fig. 1: The local conmon point

Summary

Introduction

An irreducible triangulation is a plane graph G such that (i) G is triangulated and the exterior face is a quadrangle, and (ii) G has no separating triangles (a 3-cycle with vertices both inside and outside the cycle). Given a rectangular partition, the directions of the adjacencies between rectangles can be represented as a labeling of the edges of the dual irreducible triangulation. In this labeling, all edges corresponding to horizontal The equivalence classes of the rectangular partitions with the same irreducible triangulation G as dual correspond one-to-one to the regular edge labelings of G. Our motivation to bound the number of regular edge labelings of an irreducible triangulation stems from their application to drawing rectangular cartograms. Regular edge labelings can be used to obtain straight-line drawings of these graphs on a small grid (Fusy, 2009)

Upper bound

Lower bound

Discussion