DeepAI AI Chat
Log In Sign Up

The Packing Chromatic Number of the Infinite Square Grid is 15

by   Bernardo Subercaseaux, et al.

A packing k-coloring is a natural variation on the standard notion of graph k-coloring, where vertices are assigned numbers from {1, …, k}, and any two vertices assigned a common color c ∈{1, …, k} need to be at a distance greater than c (as opposed to 1, in standard graph colorings). Despite a sequence of incremental work, determining the packing chromatic number of the infinite square grid has remained an open problem since its introduction in 2002. We culminate the search by proving this number to be 15. We achieve this result by improving the best-known method for this problem by roughly two orders of magnitude. The most important technique to boost performance is a novel, surprisingly effective propositional encoding for packing colorings. Additionally, we developed an alternative symmetry-breaking method. Since both new techniques are more complex than existing techniques for this problem, a verified approach is required to trust them. We include both techniques in a proof of unsatisfiability, reducing the trusted core to the correctness of the direct encoding.


page 4

page 16


Notes on complexity of packing coloring

A packing k-coloring for some integer k of a graph G=(V,E) is a mapping ...

S-Packing Coloring of Cubic Halin Graphs

Given a non-decreasing sequence S = (s_1, s_2, … , s_k) of positive inte...

Online bin packing of squares and cubes

In the d-dimensional online bin packing problem, d-dimensional cubes of ...

Schur Number Five

We present the solution of a century-old problem known as Schur Number F...

Reducing Moser's Square Packing Problem to a Bounded Number of Squares

The problem widely known as Moser's Square Packing Problem asks for the ...

Approximating Bin Packing with Conflict Graphs via Maximization Techniques

We give a comprehensive study of bin packing with conflicts (BPC). The i...