The 1-2-3 conjecture for polygonal tilings

Abstract

The 1-2-3 conjecture has been solved positively in 2024 for finite graphs and by extension for infinite graphs which are locally finite. The solution is non-constructive, and finding explicit solutions for large (or infinite) graphs is very hard. By exploiting the extra structure present in many non-periodic tilings, we find explicit solutions for the Chair (all three vertex placements), Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker (two versions), Penrose Rhomb, and the Domino tilings. We prove that for any fully periodic tiling of the plane there exists a fully periodic solution, and provide an algorithm for finding such a solution. We give solutions for the fully periodic square, triangle and hexagonal lattices.