# P3596: Switch diameter of one-factorizations of K_12

- ID: `P3596`
- Reference: `k12-factorization-switch-diameter`
- Page: https://theoremdb.org/statements/P3596
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/k12-factorization-switch-diameter?representation=markdown
- Record maturity: Reviewed problem

## The problem

Determine the diameter of the graph whose vertices are one-factorizations of the labeled complete graph \(K_{12}\) and whose edges are single alternating-cycle switches between two factors.

### Remarks

- **Remark.** A one-factorization partitions all edges of K_12 into 11 perfect matchings.
- **Remark.** A switch chooses two factors, takes one even cycle in their union, and exchanges the two alternating edge sets on that cycle.

### What counts as a solution

- Give the exact labeled diameter, a pair attaining it, and a complete breadth-first certificate. State separately whether quotienting vertex labels changes the answer.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

The state graph is finite, moves are cheap to replay, and frontier files prevent repeated full enumeration.

- Canonicalize only after deciding whether the target graph is labeled or isomorphism-quotiented. Store each breadth-first layer by hashes of sorted matching lists.
- Trap: switching the whole union of two factors merely swaps their names and leaves an unlabeled factorization unchanged. Proper subcycles create the useful moves.

- Recorded example: K_12 has 11!!=10395 labeled perfect matchings available as factor candidates.

### Computational notes

- Each factorization uses 11 perfect matchings and 66 edges exactly once. For any pair of factors, their union is a disjoint collection of even cycles, so all legal switches can be generated without searching arbitrary edge subsets.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `k12-factorization-switch-diameter`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.