Graph-colouring connections
Claim
The exclusion rule is an independent-set condition on a 16-vertex graph; whether that connects to anything more general is an open thread, not a result.
Construction
The badge's rule — two arrangements conflict if they share a circle or share a diamond — builds a graph on sixteen vertices with 22 edges. A legal combination is an independent set in that graph, which is what makes the count of 1,157 a combinatorial rather than a geometric fact.
On the board the same move happens in a different guise: a filled board is a perfect matching of the lattice's grid graph, and matchings are the reason the published domino-tiling sequence applies at all.
What is actually established
Two things are solid. The exclusion graph has 22 edges and its independent sets number 1,157 excluding the empty one, recomputed here. And the lattice is bipartite under the parity of i + j, which is what makes the matching argument of Claim H available.
The speculative part is the project's own: the master reference keeps a loose, unconfirmed tangent about whether the colour and adjacency structure here could connect to graph-colouring theory more broadly, or to data storage. It is recorded there as a thread not to lose rather than as active work, and that is how this page records it too.
One concrete sub-question
The exclusion graph has visible structure. Take the midpoint of each arrangement's circle-to-diamond span: the sixteen midpoints land on a closed sixteen-point star ring — five distinct radii, not a circle — with the letters running round it in order, and on that ring neighbouring arrangements always exclude one another. Only six exclusions are not ring adjacencies: B–P, C–E, E–G, H–J, K–M and M–O.
So the graph is a sixteen-cycle plus six chords, which is why it can be reasoned about by hand. Whether that structure is a coincidence of this badge or a consequence of the lattice is not answered anywhere on file.
Note
Yours. The thread is interesting; the page should not overstate it.
Open questions
- Whether the exclusion graph belongs to a named family. Nothing on file attempts the identification.
- Whether the 24-point badge's exclusion graph has the same cycle-plus-chords shape. Testable with data already in the project.
- The data-storage tangent is unexamined and should stay labelled as such.
Checks
| What | How | Source |
|---|---|---|
| Exclusion edges | 22, recomputed from the key | _src/core.py |
| Independent sets | 1,157 excluding the empty set | _src/core.py |
| Ring structure | Sixteen midpoints on a closed star ring; exactly six exclusions are not ring adjacencies | reference/geometry-constants.md |
| The tangent, as recorded | Noted as loose and unconfirmed, not active work | nuna_master_reference.md |