# Medial Annular Cycle Condition Findings

Question: for an actual plane triangulation and a tire decomposition,
does there exist a proper vertex 3-coloring of the full medial graph
such that every medial annular simple cycle uses two colors except at
at most one vertex?

The experiment is graph-level, not local-pattern-level:

1. Build the full medial graph `M(G)` from the embedding rotation.
2. Enumerate proper vertex 3-colorings of `M(G)`.
3. For each level-source tire decomposition, build the annular medial
   subgraph of each depth-component tire.
4. Enumerate simple cycles in those annular medial subgraphs.
5. Search for a proper medial coloring satisfying the almost-two-color
   condition on every such cycle.

## First Counterexample Found

Command:

```bash
sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 7 --n-max 7 --stop-on-failure --failure-details
```

Result:

```text
n=7
  FAILURE {
    'n': 7,
    'graph_idx': 0,
    'graph_edges': [
      (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7),
      (2, 3), (2, 6), (2, 7), (3, 4), (3, 5), (3, 6),
      (4, 5), (5, 6), (6, 7)
    ],
    'source': 'vertex:1',
    'cycles': 1,
    'checked': 6,
    'exhausted': True,
    'violation_size': 6,
    'violation_cycle': [
      (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7)
    ],
    'violation_counts': {0: 2, 1: 2, 2: 2},
    'violation_coloring': {
      (1, 2): 2, (1, 3): 1, (1, 4): 0,
      (1, 5): 2, (1, 6): 0, (1, 7): 1
    }
  }
```

Interpretation: for the vertex-source decomposition at source `1`, the
only annular medial simple cycle is the six medial vertices
corresponding to the six edges incident to `1`.  Sage enumerated all six
proper vertex 3-colorings of the full medial graph.  Every one violates
the almost-two-color condition on that cycle; the displayed coloring has
color counts `{0: 2, 1: 2, 2: 2}` on the annular cycle.

Thus the conjecture, in the form "for every possible tire decomposition
there exists a proper medial 3-coloring satisfying the condition on all
medial annular simple cycles", is false.

## Default Sweep

Command:

```bash
sage -python papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_medial_annular_cycle_condition.py --n-min 4 --n-max 8
```

Summary:

```text
checked source decompositions with annular cycles: 168
skipped source decompositions with no annular cycles: 0
witnesses found: 111
failures: 57
inconclusive: 0
```

Failures begin at `n=7` for vertex-source decompositions.
