bibkey: adamson2026twoword
authors: Duncan Adamson, Amanita Dietz, Pamela Fleischmann, Annika Huch, and Silas Cato Sacher
year: 2026
title: "2-word-π-representable Graphs"
doi: null
url: https://arxiv.org/abs/2605.27183v1
claim: "Definitions 1 and 14 use adjacency iff equality of two-letter projections in two k-uniform words; Conjecture 31 asks for proper adjacent inclusion at every positive k."
strata_touched:
  - D5/S1/Words/GraphRepresentation/ExplicitNonTwoUniform
  - D5/S1/Words/GraphRepresentation/UniformVertexExtension
  - D5/S1/Words/GraphRepresentation/UniformHierarchy
license: citation-only
triage: anchor

Equality of projections in two uniform words

Definition 1 starts with words w,v having the same alphabet A and defines an edge between distinct a,b precisely when their two-letter projections in the two words are equal. Definition 14 restricts both words to be k-uniform over the vertex set. At k = 2 every vertex occurs exactly twice in each word; therefore every vertex belongs to both alphabets.

The conclusion asks, without a number, for an explicit graph outside G₂. Remark 30 already establishes existence nonconstructively by a graph-class counting argument. Conjecture 31 is a different question: strictness of G_k ⊆ G_(k+1) for every k. No connectedness or minimum-size condition is attached to the explicit-graph request. Ordinary single-word alternation and unions of graphs represented by separate words are different definitions.

The candidate U₂₄ has vertex set Fin 24 ⊕ Finset (Fin 24); a point is adjacent to a subset exactly when it belongs to that subset, and there are no edges within either part. Its proposed obstruction uses four insertion cuts per subset vertex, with at most 49⁴ signatures for 2²⁴ different neighborhoods. This note attests the source question and definitions, not a published answer.

Verified locator

The source is v1, dated May 26, 2026. Definitions 1 and 14, Remark 30, Conjecture 31 and the conclusion were read from the PDF text. PDF SHA-256: f978a9e9de5cfaf8adfba59125cf16293b17f6b969f5f72594eab7a0cec719ca. The stable URL is https://arxiv.org/abs/2605.27183v1.

The supplied September 15, 2026 source-version check found only v1. The scoped exact-title and neighborhood-complexity searches found no exact published explicit witness; the latter is weak negative evidence. The candidate’s novelty is suspected-novel within that scope. Worldwide priority is ASSUMED-UNVERIFIED.

Prescribed-neighborhood extension and hierarchy boundary

Remark 2 preserves an existing representation under a common prefix or suffix. Remarks 36 and 39 insert respectively a universal or an isolated vertex. The inspected primary proof sections do not state an arbitrary-neighborhood fresh vertex extension with a one-unit uniformity increase. That construction uses old nonneighbors N and old neighbors T, each enumerated once, and the actual words x^k w xNT and x^k vNxT. Its mathematical statement is repository-derived; this source note supplies the representation definitions, not that proof.

The earlier four-author Dietz–Fleischmann–Huch–Sacher contribution in IFIG Report 2501, printed pp. 21–24 (PDF pp. 29–32), has Definition 3.1 for the same model and Conjecture 3.2 on printed p. 23 for the same all-positive-k adjacent strictness question. Its Open Problems section on p. 24 explicitly says that only k=1 strictness was proved. The actual four-page body was read from https://www.informatik.uni-giessen.de/theorietag2025/2025-Theorietag35-Schotten.pdf. This earlier statement is the same problem, not an additional settlement.

The full consumer UniformHierarchy.result proves adjacent inclusion for every positive k and every finite graph, together with a finite same-carrier separator. Its known Theorem 29 nonuniversality component is proved internally: for m=64k², the membership graph has 2^m right vertices but at most (km+1)^(2k) ≤ 2^(20k²) < 2^m ordered cut signatures. Reconstruction and injective marker cancellation force colliding signatures to have equal neighborhoods, a contradiction. This component is not a separate new target. A minimum-cardinality nonrepresented induced subset has a represented vertex deletion. Actual word transport and the frozen arbitrary-neighborhood extension give the next-uniform representation on the original subset carrier. Appending one common enumeration supplies inclusion. No literature theorem, minimality, deletion closure, relabeling or strictness is assumed as a theorem parameter. The two original left orders may differ and tied cuts are preserved.

A same-title bibliographic citation lists Gradiva, 8(9) (2022), pp. 528–533. The journal body and SSRN DOI 10.2139/ssrn.5336494 remain body-unverified; the SSRN abstract page returned HTTP 403. A title and citation match does not establish that an earlier work uses this projection-equality model or proves this extension or the hierarchy. Bounded repository, pinned Mathlib and public Lean repository searches found no exact extension supplier; the public WordRepTensor source inspected uses single-word alternation. These limits do not support a global absence or priority claim.