Add Tutte-dual bridge-derivability test; rebuild artifacts

experiments/test_tutte_bridge.py: bridge-derivability test for the dual of the 46-vertex Tutte graph (a 25-vertex non-intertwining triangulation, since the Tutte graph is non-Hamiltonian) -- the conjecture's first case beyond the n=21 Holton-McKay duals. Reuses the fast integer-state bridge engine: per source labelling with bipartite parity subgraphs, run a backward bridge-orbit BFS for an Even Level Graph witness. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
2026-05-22 12:09:13 -04:00
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5 changed files with 246 additions and 141 deletions
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+"""Bridge-derivability test for the Tutte-graph dual.
+
+We take the classic 46-vertex Tutte graph (networkx `tutte_graph`), the
+1946 counterexample to Tait's conjecture, dualize it to a 25-vertex
+triangulation D, and ask whether D -- which is NOT an intertwining tree,
+since its dual is non-Hamiltonian -- is a *bridge-derived level graph*.
+This is the conjecture's first test outside the n=21 Holton-McKay duals.
+
+Method (reuses the fast integer-state bridge engine in fast_decide):
+for each source s of D we form the BFS-parity labelling L (level mod 2),
+keep only those L for which both parity subgraphs of D are bipartite
+(a necessary condition, since bridge switches preserve bipartiteness),
+and run a backward bridge-orbit BFS looking for an Even Level Graph whose
+BFS-parity equals L. A witness proves D is bridge-derived; the BFS depth
+of the witness is the number of bridge switches from the ELG to D (the
+quantity tabulated for the n=21 duals). Not finding one within the
+state/time budget is inconclusive, not a disproof.
+"""
+import sys
+import os
+import time
+sys.path.insert(0, '/Users/didericis/Code/math-research/papers/'
+                'level_resolutions_of_maximal_planar_graphs/experiments')
+sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
+import networkx as nx
+from tutte_dual_treecolor import dual_triangulation
+from fast_bridge import EdgeCode
+from fast_decide import expand_and_check
+
+
+def characterize(D):
+    n = D.number_of_nodes()
+    m = D.number_of_edges()
+    is_tri = (m == 3 * n - 6)
+    planar, _ = nx.check_planarity(D)
+    # A triangulation is 4-connected iff it has no separating triangle,
+    # i.e. iff #triangles == #faces == 2n-4. Extra triangles are separating
+    # (dually: the cubic graph has a non-trivial 3-cut -> cyclically
+    # 3-connected, not 5-connected).
+    n_triangles = sum(nx.triangles(D).values()) // 3
+    faces = 2 * n - 4
+    sep_triangles = n_triangles - faces
+    return {
+        'n': n, 'm': m, 'is_triangulation': is_tri, 'planar': planar,
+        'triangles': n_triangles, 'faces': faces,
+        'separating_triangles': sep_triangles,
+        'four_connected': sep_triangles == 0,
+    }
+
+
+def valid_parity_partitions_via_coloring(D, max_partitions=400,
+                                         max_colorings=2_000_000):
+    """A partition V = even u odd with both parity subgraphs bipartite is
+    exactly a proper 4-coloring of D grouped {0,1}|{2,3} (each parity class
+    is then 2-colored, hence bipartite). Enumerate proper 4-colorings by
+    backtracking (fixing colour of the first vertex to break the 4! colour
+    symmetry only partially) and collect the distinct induced partitions."""
+    nodes = sorted(D.nodes())
+    adj = {v: set(D.neighbors(v)) for v in nodes}
+    # colour high-degree vertices first -> heavier pruning
+    order = sorted(nodes, key=lambda v: -len(adj[v]))
+    colors = {}
+    parts = {}        # canonical partition key -> labels dict
+    n_col = [0]
+
+    def record():
+        n_col[0] += 1
+        even = frozenset(v for v in nodes if colors[v] in (0, 1))
+        odd = frozenset(nodes) - even
+        if not odd:
+            return
+        key = min((even, odd), key=lambda s: (len(s), tuple(sorted(s))))
+        if key not in parts:
+            parts[key] = {v: (0 if v in even else 1) for v in nodes}
+
+    def bt(i):
+        if len(parts) >= max_partitions or n_col[0] >= max_colorings:
+            return
+        if i == len(order):
+            record()
+            return
+        v = order[i]
+        used = {colors[u] for u in adj[v] if u in colors}
+        cap = 1 if i == 0 else 4   # fix first vertex's colour to 0
+        for c in range(cap):
+            if c in used:
+                continue
+            colors[v] = c
+            bt(i + 1)
+            del colors[v]
+            if len(parts) >= max_partitions or n_col[0] >= max_colorings:
+                return
+
+    bt(0)
+    return list(parts.values()), n_col[0]
+
+
+def search_partition(code, labels, n, cap, time_limit):
+    """Backward bridge-orbit BFS tracking depth. Returns
+    (status, n_states, depth_of_witness_or_None)."""
+    s0 = code.state0
+    seen = {s0}
+    frontier = [s0]
+    depth = 0
+    t0 = time.time()
+    while frontier and len(seen) < cap:
+        if time.time() - t0 > time_limit:
+            return 'timeout', len(seen), None
+        new = []
+        for st in frontier:
+            wit, neigh = expand_and_check(st, code, labels, n)
+            if wit:
+                return 'found', len(seen), depth
+            for ns in neigh:
+                if ns not in seen:
+                    seen.add(ns)
+                    new.append(ns)
+                    if len(seen) >= cap:
+                        break
+            if len(seen) >= cap:
+                break
+        frontier = new
+        depth += 1
+    return ('exhausted' if not frontier else 'capped'), len(seen), None
+
+
+def main(cap=400_000, time_limit=120.0):
+    G = nx.tutte_graph()
+    D, _ = dual_triangulation(G)
+    info = characterize(D)
+    print('Tutte graph: |V|=%d (classic 46-vertex Tait counterexample)'
+          % G.number_of_nodes())
+    print('Dual D: n=%(n)d, m=%(m)d, triangulation=%(is_triangulation)s, '
+          'planar=%(planar)s' % info)
+    print('  triangles=%(triangles)d, faces=%(faces)d, '
+          'separating triangles=%(separating_triangles)d, '
+          '4-connected=%(four_connected)s' % info)
+    print('  [non-intertwining-tree by Thm: dual of a non-Hamiltonian C3CP]')
+
+    code = EdgeCode(D.nodes())
+    code.state0 = code.state_of(D)
+    n = info['n']
+
+    parts, n_col = valid_parity_partitions_via_coloring(D)
+    print('\n%d distinct valid parity partitions collected '
+          '(from %d proper 4-colorings)' % (len(parts), n_col))
+    if not parts:
+        print('No valid parity partition found: D is not bridge-derivable.')
+        return
+
+    t0 = time.time()
+    for k, labels in enumerate(parts):
+        ev = sum(1 for v in labels if labels[v] == 0)
+        st, sz, depth = search_partition(code, labels, n, cap, time_limit)
+        tag = {'found': 'WITNESS', 'exhausted': 'exhausted (no witness)',
+               'capped': 'CAP hit', 'timeout': 'TIMEOUT'}[st]
+        if st == 'found' or (k + 1) % 20 == 0 or st in ('capped', 'timeout'):
+            print('  partition %3d (even=%2d odd=%2d): %s  [orbit>=%d, %.0fs]'
+                  % (k, ev, n - ev, tag, sz, time.time() - t0))
+        if st == 'found':
+            print('\n==> Tutte dual IS a bridge-derived level graph: '
+                  'ELG witness at %d bridge switches (partition %d).'
+                  % (depth, k))
+            return
+    print('\n==> No ELG witness found over %d valid partitions within budget '
+          '(cap=%d, %.0fs/partition). INCONCLUSIVE.'
+          % (len(parts), cap, time_limit))
+
+
+if __name__ == '__main__':
+    main()
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