Post #2400404
2026-04-14 15:05 UTC
@simontatham@hachyderm.io dumb question: P3 is related to tesseract tilings of 5-space, so do the 2-cells of the tiling of 5-space admit a 4-colouring? It feels like the answer should be obviously yes or obviously no but I'm rusty at this.
Replies (1)
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@simontatham@hachyderm.io 2026-04-14 15:38
@petermilley@zeroes.ca I think no, because even the 2-cells of a single 5-hypercube don't admit a 4-colouring. At any vertex V, ten squares meet, each corresponding to a different pair of the 5 edges leaving V. Two squares meet if they share an edge. So the adjacency graph between those 10 squares is the complement of the Petersen graph (which connects 2-subsets of [5] iff they are disjoint). That graph has chromatic number 5. So you need five colours just to colour all the square faces meeting at one vertex of the hypercube, let alone the whole hypercube, let alone the whole hypercube lattice. So this only works for a P3 tiling because it's a small subset of the 2-cells of the lattice.