A packing $k$-coloring is a natural variation on the standard notion of graph $k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and any two vertices assigned a common color $c \in \{1, \ldots, k\}$ 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.
翻译:包装 $k$ 彩色 是 美元 $k$ 彩色 标准概念 的自然变异 。 以 $1,\ ldots, k ⁇ $, 和 任何两只脊椎为 $1,\\ $1,\ ldots, k ⁇ $ 分配的通用彩色, 需要距离大于 $c 美元( 而不是 美元, 标准图形颜色 ) 。 尽管有一系列渐进式的工作, 确定无限平方格的包装色素数自2002年推出以来仍然是一个未解决的问题 。 我们通过证明这个数字为15, 最终通过改进最著名的解决这个问题的方法, 大约用两个数量级来达到这个结果 。 最重要的提高性能的方法是新颖的、 出奇异的包装彩色的理论编码 。 此外, 我们开发了一种替代的对质断法方法。 由于这两种新技术都比 对这个问题的现有技术更为复杂,, 需要一种经过验证的方法来信任它们。 我们把这两种技术都包括在不满足性的证据中,,, 将信任的核心 降低 直接 正确 。