The 1-2-3 conjecture for polygonal tilings

TL;DR

This paper extends the 1-2-3 conjecture to polygonal tilings, providing explicit solutions for various non-periodic and periodic tilings, and offers algorithms for finding such solutions.

Contribution

It introduces explicit solutions for the 1-2-3 conjecture in polygonal tilings, including non-periodic and periodic cases, with algorithms for periodic tilings.

Findings

Explicit solutions for Chair, Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker, Penrose Rhomb, and Domino tilings.

Existence of fully periodic solutions for any fully periodic tiling of the plane.

Algorithms provided for finding solutions in fully periodic square, triangle, and hexagonal lattices.

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.