face_monochromatic_pairs: strengthen Conj 5.5 to face-length ≥ 5, find 28-vertex counterexample

- Paper: Conjecture 5.5 restated with the hypothesis that every face
  of H has length ≥ 5 (= the cubic plane analogue of "no triangles
  or quadrilaterals as faces"). This kills K_4 and the n=8 trivial
  counterexamples (girth 3) and the ad-hoc n=40 counterexample
  (which has 2 triangles and 4 quadrilaterals). A new remark catalogues
  these excluded counterexamples and the smallest cubic plane graphs
  satisfying the hypothesis (dodecahedron at |V| = 20).
- search_smaller_counterexample.py: --min-face=N option to filter
  cubic planar graphs by minimum face length.
- search_min_face5_counterexample.py: enumerates triangulations T
  with min degree ≥ 5 via graphs.triangulations(n, minimum_degree=5),
  takes planar dual (= cubic plane with all faces ≥ 5), and runs the
  Heawood-constancy check.
- Result: smallest counterexample at triangulation order n_T = 16,
  whose dual is a 28-vertex cubic plane graph (graph6
  [kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a???W???A_???F).
  Faces: 12 pentagons + 4 hexagons (a C28 fullerene). Both
  K_{red, blue} and K_{red, green} are 12-cycles sharing the
  colour-red edge (0, 1) and both have h_φ ≡ -1. 8 of 28 vertices
  lie outside V(K_0) ∪ V(K_1).
- verify_28_vertex_counterexample.py: reproduces the counterexample,
  verifies all properties, and renders figures/min-face-5-counterexample.png.

Note on the boundary: face-length ≥ 6 is impossible for cubic plane
graphs by Euler (6F = 6(V/2 + 2) > 3V = sum face lengths for V > 4).
So face-length ≥ 5 is the strongest face-length restriction admitting
any cubic plane graphs at all.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/face_monochromatic_pairs/experiments/search_smaller_counterexample.py

@@ -146,19 +146,45 @@ def check_graph(G):
     return None
 
 
+def min_face_length(G):
+    """Min face length of G in its current planar embedding."""
+    return min(len(f) for f in G.faces())
+
+
 def main():
     args = [a for a in sys.argv[1:] if not a.startswith('--')]
     max_n = int(args[0]) if args else 18
     flag_all = '--all' in sys.argv[1:]
+    # Parse --min-face N
+    min_face = None
+    for i, a in enumerate(sys.argv[1:]):
+        if a == '--min-face' and i + 1 < len(sys.argv) - 1:
+            min_face = int(sys.argv[i + 2])
+            break
+        if a.startswith('--min-face='):
+            min_face = int(a.split('=', 1)[1])
+            break
 
     print(f"Searching for cubic plane graph counterexamples to "
           f"Conjecture 5.5, n in [4, {max_n}] "
-          f"({'continuing past first hit' if flag_all else 'stopping at first hit'})\n")
+          f"({'continuing past first hit' if flag_all else 'stopping at first hit'})"
+          + (f", min face length >= {min_face}" if min_face else "")
+          + "\n")
+
+    # If filtering by min face length, skip small n where impossible.
+    # For all-faces-length->=L cubic plane: V - E + F = 2, E = 3V/2,
+    # sum face lengths = 3V, F = V/2 + 2, so min sum = L*(V/2 + 2)
+    # <= 3V gives V >= 2L*(L-3)/(3-L/2)... let's just set V >= 20 for L=5.
+    if min_face is not None and min_face >= 5:
+        n_start = max(4, 20)
+    else:
+        n_start = 4
 
     first_found = None
-    for n in range(4, max_n + 1, 2):   # cubic requires n even
+    for n in range(n_start, max_n + 1, 2):   # cubic requires n even
         start = time.time()
         count = 0
+        skipped = 0
         found = None
         try:
             gen = graphs.planar_graphs(
@@ -172,6 +198,11 @@ def main():
         for G in gen:
             if max(G.degree()) != 3:
                 continue   # not cubic
+            if min_face is not None:
+                G.is_planar(set_embedding=True)
+                if min_face_length(G) < min_face:
+                    skipped += 1
+                    continue
             count += 1
             res = check_graph(G)
             if res is not None:
@@ -180,8 +211,10 @@ def main():
                     break
         elapsed = time.time() - start
         if found is None:
-            print(f"n={n:>3}: checked {count} graphs, no counterexample "
-                  f"[{elapsed:.1f}s]")
+            extra = (f", skipped {skipped} due to small face"
+                     if min_face is not None else "")
+            print(f"n={n:>3}: checked {count} graphs, no counterexample"
+                  f"{extra} [{elapsed:.1f}s]")
         else:
             G, (col, K0, K1, h0, h1, a, b, c, e) = found
             colour_name = {0: 'red', 1: 'blue', 2: 'green'}