Virtual-knot problem 55 — Construct a parity on graph-links by using Bouchet's criterion about the realizability of a graph.

Definitions and mathematical statement

A graph is realizable as a chord-intersection graph when it can be obtained from chords on a circle, with two vertices adjacent exactly when the endpoints of the corresponding chords alternate around the circle. This is a usual meaning of realizability in the graph-link setting, but the source does not specify which version of Bouchet's criterion it intends.

Graph-links are equivalence classes of graph diagrams under specified graph Reidemeister-type moves, with any vertex labels or framings required by the source. The record gives neither those moves nor their label conventions. A parity would assign labels, commonly in ${\mathbb{Z}}/2{\mathbb{Z}}$, to graph vertices and satisfy the corresponding move-compatibility conditions. Those conditions cannot be inferred merely from realizability of the underlying unlabelled graph.

Problem-list item 55 asks to construct a parity on graph-links using Bouchet's realizability criterion. It further asks for a parity "responsible for the cyclically 6-edge connection." In a common graph-theoretic convention a graph is cyclically $6$-edge-connected when no edge cut of size less than $6$ separates it into pieces each containing a cycle. This is only a conditional explanation: the source does not define the phrase, identify which associated graph it concerns, or specify what responsibility of a parity means. No equivalence, obstruction theorem, or nontriviality condition is supplied. A precise construction must recover the graph-link move system, Bouchet criterion and intended connection rather than substitute an arbitrary realizability label as a parity.

Short English statement

Build a graph-link parity from the intended Bouchet graph-realizability criterion, and relate it to the source's cyclic $6$-edge-connectivity phenomenon. The required move compatibility and the meaning of that relation are missing, along with the in-statement reference.

Sources

• Local UnsolvedMath registry, permanent ID $10600055$, exact source code catalogue entry, virtual-knot problem-list item 55. The statement names Bouchet's criterion but has an empty citation after "see." No individual proposer or proposed year is supplied.
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, dedicated problem-list item 55: https://arxiv.org/abs/1409.2823. Background: 60 top-level items, compound items retained intact; source-TeX extraction, accessed 2026-07-29.
• Supplied research summary mentions Bouchet's interlacement/realizability framework and interlace polynomials, with incomplete identifiers arXiv:0209.xxxx (2002) and arXiv:0606.xxxx (2006). No titles or complete references are provided, and the identifiers are not completed here.
• Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied OPEN-TRIAGE assessment reports no literature found on this particular application and no resolution; that is a supplied search assessment, not independently verified absence.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.