Test inner-boundary conjecture on Holton-McKay duals

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
2026-06-01 20:56:19 -04:00
parent 88c74efd28
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+"""Test the tire inner-boundary three-colour conjecture
+(Conjecture 1.31, conj:tire-inner-boundary-three-colour) on the six
+duals of the non-Hamiltonian 38-vertex cubic plane graphs found by
+Holton & McKay.
+
+Each Holton-McKay graph G' is a 3-connected cubic plane graph on 38
+vertices; its planar dual G is a 21-vertex triangulation.  We run the
+exists-source, vertex-rooted inner-boundary check on each dual.
+
+Run with:
+    sage -python experiments/check_inner_boundary_on_holton_mckay.py
+"""
+from __future__ import annotations
+
+import os
+import sys
+from typing import cast
+
+from sage.all import Graph  # type: ignore[attr-defined]  # pylint: disable=no-name-in-module
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+
+from check_level_cycle_three_color import (  # noqa: E402
+    level_sources,
+    test_graph,
+)
+
+
+HM_FILE = (
+    "/Users/didericis/Code/math-research/papers/"
+    "even_level_graph_generators/experiments/nonham38m4.pc"
+)
+
+
+def parse_planar_code_to_sage(path: str) -> list[Graph]:
+    """Parse McKay's planar_code file -> list of Sage Graphs."""
+    with open(path, "rb") as f:
+        data = f.read()
+    header = b">>planar_code<<"
+    assert data.startswith(header), data[:20]
+    pos = len(header)
+    out: list[Graph] = []
+    while pos < len(data):
+        n = data[pos]
+        pos += 1
+        edges = set()
+        for v in range(n):
+            while True:
+                w = data[pos]
+                pos += 1
+                if w == 0:
+                    break
+                edges.add(frozenset((v, w - 1)))
+        g = Graph([tuple(e) for e in edges], multiedges=False, loops=False)
+        g.is_planar(set_embedding=True)
+        out.append(g)
+    return out
+
+
+def main() -> int:
+    hm = parse_planar_code_to_sage(HM_FILE)
+    print(f"Loaded {len(hm)} Holton-McKay graphs from {HM_FILE}\n")
+
+    all_pass = True
+    for i, gprime in enumerate(hm, start=1):
+        # Dual: triangulation of the sphere.  Sage's planar_dual requires a
+        # plane embedding; gprime already has one set above.
+        dual = cast(Graph, gprime.planar_dual())
+        n = dual.order()
+        e = dual.size()
+        print(f"=== HM #{i}: G' has {gprime.order()} vertices, "
+              f"{gprime.size()} edges; dual has n={n}, m={e} ===")
+        # Sanity: triangulation has 3n - 6 edges.
+        if e != 3 * n - 6:
+            print(f"  WARNING: dual is not a triangulation (m != 3n-6)")
+
+        sources = list(level_sources(dual, "vertex", None))
+        passed, complete, checked = test_graph(
+            dual,
+            sources,
+            max_colorings=None,
+            stop_first=True,
+            quantifier="exists-source",
+            restriction="inner-boundary",
+        )
+        status = "PASS" if (passed and complete) else (
+            "UNKNOWN" if not complete else "FAIL"
+        )
+        print(f"  {status}: sources_tried={checked}/{len(sources)}\n")
+        if not (passed and complete):
+            all_pass = False
+
+    print("---")
+    print(
+        "All Holton-McKay duals satisfy the inner-boundary conjecture: "
+        f"{all_pass}"
+    )
+    return 0 if all_pass else 1
+
+
+if __name__ == "__main__":
+    raise SystemExit(main())
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@@ -1540,6 +1540,13 @@ conjecture.}
 \label{tab:inner-boundary-three-colour-c5}
 \end{table}

+We also re-ran the inner-boundary check on the six dual triangulations
+of the Holton--McKay graphs and found a vertex source witnessing the
+conjecture for each.  In four of the six cases the first source tried
+already succeeded; in the remaining two, one or two vertex sources
+exhausted all $4320$ proper $4$-colourings before a witnessing source
+was found.
+
 Unlike the small-$n$ census, where the first source and colouring tried
 typically already witness the restriction, the source choice is
 genuinely active in the $5$-connected slice: many vertex sources fail