Coloring Unit-Distance Strips using SAT

17 pages•Published: May 27, 2020

Peter Oostema, Ruben Martins and Marijn Heule

Abstract

Satisfiability (SAT) solving has become an important technology in computer-aided mathematics with various successes in number and graph theory. In this paper we apply SAT solvers to color infinitely long strips in the plane with a given height and number of colors. The coloring is constrained as follows: two points that are exactly unit distance apart must be colored differently. To finitize the problem, we tile the strips and all points on a tile have the same color. We evaluated our approach using two different tile shapes: squares and hexagons. The visualization of bounded height strips using 3 to 6 colors reveal patterns that are similar to the best known lower bounds for infinite strips. Our method can be a useful tool for mathematicians to search for patterns that can be generalized to infinite strips and allowed us to increase the lower bound for the strip height with 5 colors to an improved height of 1.700084.

Keyphrases: chromatic number of the plane, graph coloring, sat solving

In: Elvira Albert and Laura Kovacs (editors). LPAR23. LPAR-23: 23rd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 73, pages 373-389.

Links:	https://easychair.org/publications/paper/69T4
	https://doi.org/10.29007/btmj

BibTeX entry

@inproceedings{LPAR23:Coloring_Unit_Distance_Strips,
  author    = {Peter Oostema and Ruben Martins and Marijn Heule},
  title     = {Coloring Unit-Distance Strips using SAT},
  booktitle = {LPAR23. LPAR-23: 23rd International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Elvira Albert and Laura Kovacs},
  series    = {EPiC Series in Computing},
  volume    = {73},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/69T4},
  doi       = {10.29007/btmj},
  pages     = {373-389},
  year      = {2020}}

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