Replace counterexample with minimal n=13 graph found by exhaustive search

Adds search_counterexample_comprehensive iterating Sage's planar_graphs
generator across all maximal planar graphs of bounded order. Exhaustive
enumeration through order 13 (9150+49566 triangulations) yields exactly
one graph with no plane diamond coloring, at order 13. Updates Theorem
2.6 to assert minimality and uniqueness, and replaces the figure and
edge list with the smaller counterexample.

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

plane_diamond_coloring.py

@@ -62,13 +62,40 @@ def search_counterexample(n: int, num_trials: int) -> Graph | None:
     return None
 
 
+def search_counterexample_comprehensive(max_order: int, min_order: int = 4) -> list[Graph]:
+    """
+    Iterate through every maximal planar graph of order in [min_order, max_order]
+    and return all those without a plane diamond coloring.
+    """
+    counterexamples: list[Graph] = []
+    for n in range(min_order, max_order + 1):
+        checked = 0
+        for g in graphs.planar_graphs(n, minimum_connectivity=3, maximum_face_size=3):
+            checked += 1
+            if not has_plane_diamond_coloring(g):
+                print(f"Counterexample at order {n} (graph #{checked}): {g.graph6_string()}")
+                counterexamples.append(g)
+            if checked % 100 == 0:
+                print(f"  order {n}: checked {checked} graphs, {len(counterexamples)} counterexamples so far")
+        print(f"order {n} done: {checked} triangulations checked")
+    return counterexamples
+
+
 if __name__ == "__main__":
     import sys
-    n = int(sys.argv[1]) if len(sys.argv) > 1 else 12
-    num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100
-    counterexample = search_counterexample(n, num_trials)
-    if counterexample is None:
-        print(f"No counterexample found in {num_trials} random triangulations of order {n}")
+    if len(sys.argv) > 1 and sys.argv[1] == "comprehensive":
+        max_order = int(sys.argv[2]) if len(sys.argv) > 2 else 10
+        min_order = int(sys.argv[3]) if len(sys.argv) > 3 else 4
+        counterexamples = search_counterexample_comprehensive(max_order, min_order)
+        print(f"Found {len(counterexamples)} counterexamples in orders {min_order}..{max_order}")
+        for g in counterexamples:
+            canonize_and_save_graph(g)
     else:
-        canonical, graph_dir = canonize_and_save_graph(counterexample)
-        print(f"Counterexample saved to {graph_dir}")
+        n = int(sys.argv[1]) if len(sys.argv) > 1 else 12
+        num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100
+        counterexample = search_counterexample(n, num_trials)
+        if counterexample is None:
+            print(f"No counterexample found in {num_trials} random triangulations of order {n}")
+        else:
+            canonical, graph_dir = canonize_and_save_graph(counterexample)
+            print(f"Counterexample saved to {graph_dir}")