Happy Edges: Threshold-Coloring of Regular Lattices

TL;DR

This paper investigates a graph coloring problem inspired by a puzzle, focusing on threshold-coloring of Archimedean and Laves lattices, revealing which are colorable with bounded colors, unbounded colors, or not at all.

Contribution

It characterizes threshold-colorability of regular lattice tilings, identifying conditions for bounded, unbounded, or impossible colorings under various edge labelings.

Findings

Some lattices are threshold-colorable with a constant number of colors.

Certain labelings require an unbounded number of colors.

Some lattices are not threshold-colorable at all.

Abstract

We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into {\em near} and {\em far} sets and an integer threshold $t$, a {\em threshold-coloring} of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by $t$ or less, while endpoints of far edges differ by more than $t$. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.