Distances on lozenge tilings

Authors:
Olivier Bodini;Thomas Fernique;Éric Rémila
Affiliations:
LIP6, CNRS & Univ. Paris 6, Paris, France;LIF, CNRS & Univ. de Provence, Marseille, France;LIP, CNRS & Univ. de Lyon 1, Lyon, France
Venue:
DGCI'09 Proceedings of the 15th IAPR international conference on Discrete geometry for computer imagery
Year:
2009

Citing 5
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Abstract

In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the the number of flips - a local transformation on tilings - necessary to link two tilings. We prove that the flip-distance between two tilings is equal to the Hamming-distance for n ≤ 4. We also show, by providing a pair of so-called deficient tilings, that this does not hold for n ≥ 6. We finally discuss the n = 5 case, which remains open.