bibkey: erickson2024metamour
authors: William Q. Erickson, Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, Mitchell Minyard, Kyle Rosengartner
year: 2024
title: "Limits and Periodicity of Metamour 2-Distance Graphs"
doi: null
url: https://arxiv.org/abs/2409.02306v1
claim: "Theorem 4.1 characterizes fixed points of the metamour operator as edgeless graphs; the proof of Theorem 6.5 explicitly states that K_1 has metamour period one."
strata_touched:
  - D5/S3/Combinatorics/Graph/Walks/DemoStrongPeriodBoundRefutation
license: citation-only
triage: anchor

Known fixed points of the exact-distance-two operator

Verified locator

The fixed source is arXiv:2409.02306v1, original PDF and original HTML.

The complete thirty-four-page original PDF and relevant HTML were inspected in the source qualification. Definition 2.1 gives the metamour graph by adjacency at exact shortest distance two. Theorem 4.1 states that a graph is a fixed point exactly when it is edgeless. The proof of Theorem 6.5 explicitly notes that K_1 has metamour period one while excluding it from the period-two case.

These are prior literature facts. In particular the connected singleton and its period one are not new. They support the elementary instance used against the later conjecture in demo2026periodicity; this source does not state a refutation of that later bound. The Lean proof uses the actual distance and identity-isomorphism facts directly, without assuming this paper’s theorems or adding a separately packaged fixed-point theorem.