> At this point, we were conjecturing that using color in the chessboard pattern that yields density 1/2 was optimal, meaning that you could packing-color a graph D_r with k colors if, and only if, you could do it enforcing the chessboard pattern of 1's.
> After more optimizations, it turned out the chessboard conjecture was false
> Figure 19: The smallest counterexample to the chessboard conjecture. The diamond of radius 14, and a 6 forced in the center, can be packing-colored with 14 colors only if the chessboard pattern is broken.
I don't understand why this is a counterexample to the conjecture. Painting the center with color 6 isn't part of the graph. Can the diamond of radius 14 be colored using a checkerboard pattern of 1s when the center color isn't 6?
Is the diamond of radius 14, with a 2 forced in the center and in all 4 squares adjacent to the center, a counterexample to the conjecture that the infinite grid is colorable at all?
> After more optimizations, it turned out the chessboard conjecture was false
> Figure 19: The smallest counterexample to the chessboard conjecture. The diamond of radius 14, and a 6 forced in the center, can be packing-colored with 14 colors only if the chessboard pattern is broken.
I don't understand why this is a counterexample to the conjecture. Painting the center with color 6 isn't part of the graph. Can the diamond of radius 14 be colored using a checkerboard pattern of 1s when the center color isn't 6?
Is the diamond of radius 14, with a 2 forced in the center and in all 4 squares adjacent to the center, a counterexample to the conjecture that the infinite grid is colorable at all?