The $S$-packing coloring of the infinite diagonal grid with $S = (1,6,6,\ldots)$

Abstract

For a non-decreasing sequence of positive integers $S = (a_1, a_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is the smallest positive integer $k$ such that the vertices can be colored with $k$ colors, where the distance between any two distinct vertices of color $i$ is greater than $a_i$. In this paper, we show that the $S$-packing chromatic number of the infinite diagonal grid $P_\infty \boxtimes P_\infty$ with $S = (1,6,6,\ldots)$ is $40$. This confirms a conjecture of the first author and Tiyajamorn.