bibkey: dutta2026a396659
authors: Arkadeep Dutta
year: 2026
title: "OEIS A396659: edge-collision walks visiting every vertex"
doi: null
url: https://oeis.org/A396659/internal
claim: "The FORMULA conjecture asks whether T(n,n^2)/T(n,n^2+1)=(2n-1)/n for every n at least two, with a fixed starting vertex and ordered directed edges including loops."
strata_touched:
  - D5/S3/Combinatorics/Graph/Walks/CollisionPrefix
license: citation-only
triage: anchor

Edge-collision walks on a complete directed graph

Source and conventions

OEIS A396659, revision 16, defines T(n,k) using n labeled vertices and a fixed starting vertex. Every ordered pair of vertices is an available directed edge, including self-loops. A qualifying walk visits every vertex, uses pairwise distinct first k-1 edges, and repeats one of those edges at its final step. The walks are ordered sequences with a distinguished first position.

The ratio conjecture quantifies over every n at least two. The same entry separately asserts a factorial formula for T(n,n^2+1). The ratio argument under consideration does not assume that formula. A source label of conjecture and the bounded source checks do not establish worldwide openness or priority.

Mathematical boundary

The ordered-edge and vertex-function carriers, the two possible omitted-edge configurations of an almost-complete trail, and the append/delete correspondence for the denominator are specified in CollisionPrefix. The same module supplies the marked loop-insertion bijection, both exact cardinality identities, constructive Euler nonemptiness, a positive source denominator and the all-n rational ratio. These results do not assume the entry’s separate last-column factorial formula. Canonical freeze is retained. Current source/form and retained-Reg validation, independent delivery review, required CI and merged delivery remain unfinished. All eight public-theorem escape audits remain unfinished under the linked-issue exception.

Verified locator

https://oeis.org/A396659/internal