Verify Remark 5.8 on genuine bite treads

Bites arise when the inner outerplanar graph O has a bridge: the bridge
edge is traversed twice by the outer-face walk, so its medial vertex is
adjacent to four annular vertices.

- check_remark58_bite.py: a minimal bite tread (outer 4-cycle + interior
  bridge u-w) restricts to Kempe-balanced on all colourings (outer face).
- check_remark58_bite_rich.py: O = triangle abc + pendant bridge a-d gives
  one bite plus three singleton down teeth in the bite's inner-gap face;
  every restriction is Kempe-balanced (the three gap singletons are a
  rainbow in every global colouring).

Update Remark 5.8's verification note: the bite case, including singletons
in the bite-gap face, is now confirmed.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_remark58_bite.py

@@ -0,0 +1,116 @@
+"""Directly test Remark 5.8 on a genuine tire piece that contains a BITE.
+
+A bite arises when the inner outerplanar graph O has a bridge: the bridge edge
+is traversed twice by the outer-face walk, so it borders two tread triangles and
+its medial vertex is adjacent to four annular medial vertices.
+
+Minimal construction.  Outer 4-cycle o0,o1,o2,o3; two interior vertices u,w
+joined by a bridge u-w (V_in = {u,w}).  Triangulate the disk so that u-w lies in
+two tread triangles:
+
+    (o0,o1,u) (o0,u,o3) (o1,w,u) (o1,o2,w) (o2,o3,w) (o3,u,w)
+
+Cap the outer cycle with an apex N (the bridge bounds no inner hole, so no inner
+cap is needed).  The result G is a closed plane triangulation; M(G) is 4-regular.
+
+Edge classification (by endpoints): annular = one endpoint outer & one inner;
+up tooth = both endpoints outer (outer-cycle edge); down tooth = both endpoints
+inner (here only the bridge u-w).  The bridge's medial vertex is the bite apex.
+
+Remark 5.8 predicts every proper 3-colouring of M(G) restricts to a
+Kempe-balanced colouring.  Here the only non-trivial condition is the outer face
+(the four up apexes), since the single bite contributes no singleton down teeth.
+"""
+
+from __future__ import annotations
+
+import networkx as nx
+
+from check_remark58_bitefree import ekey, medial_graph, proper_3_colorings
+
+PAIRS = ((0, 1), (0, 2), (1, 2))
+
+OUTER = ["o0", "o1", "o2", "o3"]
+INNER = ["u", "w"]
+
+TREAD_TRIANGLES = [
+    ("o0", "o1", "u"),
+    ("o0", "u", "o3"),
+    ("o1", "w", "u"),
+    ("o1", "o2", "w"),
+    ("o2", "o3", "w"),
+    ("o3", "u", "w"),
+]
+
+
+def build():
+    g = nx.Graph()
+    for tri in TREAD_TRIANGLES:
+        a, b, c = tri
+        g.add_edges_from([(a, b), (b, c), (a, c)])
+    # outer cap
+    for i in range(4):
+        g.add_edges_from([("N", OUTER[i]), ("N", OUTER[(i + 1) % 4])])
+    return g
+
+
+def classify_tread_edges(g):
+    out = set(OUTER)
+    inn = set(INNER)
+    tread_edges = set()
+    for tri in TREAD_TRIANGLES:
+        a, b, c = tri
+        tread_edges |= {ekey(a, b), ekey(b, c), ekey(a, c)}
+    annular, up, down = [], [], []
+    for e in tread_edges:
+        a, b = e
+        ao, bo = a in out, b in out
+        ai, bi = a in inn, b in inn
+        if (ao and bi) or (ai and bo):
+            annular.append(e)
+        elif ao and bo:
+            up.append(e)
+        elif ai and bi:
+            down.append(e)
+    return annular, up, down
+
+
+def run():
+    g = build()
+    assert nx.check_planarity(g)[0]
+    M = medial_graph(g)
+    annular, up, down = classify_tread_edges(g)
+    annular_set = set(annular)
+
+    # confirm there is a bite: a down edge whose medial vertex has 4 annular nbrs
+    bites = [e for e in down if sum(1 for nb in M.neighbors(e) if nb in annular_set) == 4]
+    print(f"tread: annular={len(annular)} up={len(up)} down={len(down)} "
+          f"bite apexes={len(bites)} (bite edge: {bites})")
+
+    colorings = proper_3_colorings(M, limit=20000)
+    balanced = 0
+    bad = []
+    for col in colorings:
+        ok = all(
+            sum(1 for e in up if col[e] in pair) % 2 == 0
+            for pair in PAIRS
+        )
+        if ok:
+            balanced += 1
+        else:
+            bad.append(col)
+
+    print(f"|V(G)|={g.number_of_nodes()} |M(G)|={M.number_of_nodes()} "
+          f"colourings tested={len(colorings)}")
+    print(f"    outer-face (up-apex) balanced={balanced}  UNBALANCED={len(bad)}")
+    if bad:
+        print(f"    first unbalanced up colours: {[bad[0][e] for e in up]}")
+    print()
+    print("Remark 5.8 holds on this bite tread"
+          if not bad else
+          "Remark 5.8 FAILS on this bite tread")
+    return len(bad)
+
+
+if __name__ == "__main__":
+    run()

papers/medial_tire_decompositions_of_plane_triangulations/experiments/check_remark58_bite_rich.py

@@ -0,0 +1,132 @@
+"""Test Remark 5.8 on a bite tread that also has singleton down teeth in the
+bite's inner-gap face -- the subtle case of the condition.
+
+Inner outerplanar graph O = triangle (a,b,c) plus a pendant bridge a-d.  Its
+outer-face walk is the cyclic sequence W = [d, a, b, c, a]: the bridge a-d is
+traversed twice (-> a bite), the triangle edges a-b, b-c, c-a once each (-> three
+singleton down teeth, all sitting in the bite's inner-gap face).
+
+We triangulate the annulus between an outer m-cycle and W by the lattice-path
+method, searching interleavings for one giving a simple closed triangulation
+after capping the outer cycle with an apex N.  Then we test Remark 5.8: every
+proper 3-colouring of M(G) restricts to a Kempe-balanced colouring, i.e.
+
+  * the up apexes (outer edges) are even per colour pair, and
+  * the three singleton down apexes (a-b, b-c, c-a), which share the bite-gap
+    face, are even per colour pair (equivalently: a rainbow).
+"""
+
+from __future__ import annotations
+
+import itertools
+
+import networkx as nx
+
+from check_remark58_bitefree import ekey, medial_graph, proper_3_colorings
+
+PAIRS = ((0, 1), (0, 2), (1, 2))
+INNER_WALK = ["d", "a", "b", "c", "a"]   # bridge a-d traversed twice
+SINGLETON_DOWN = [ekey("a", "b"), ekey("b", "c"), ekey("c", "a")]
+BITE_EDGE = ekey("a", "d")
+
+
+def build_tread(m: int, path: str):
+    """Build the annular triangulation for a given lattice path (m 'O', L 'I')."""
+    outer = [f"o{t}" for t in range(m)]
+    W = INNER_WALK
+    L = len(W)
+    g = nx.Graph()
+    g.add_edge(outer[0], W[0])           # anchor
+    i = j = 0
+    tread_triangles = []
+    for mv in path:
+        if mv == "O":
+            tri = (outer[i % m], W[j % L], outer[(i + 1) % m])
+            i += 1
+        else:
+            tri = (outer[i % m], W[j % L], W[(j + 1) % L])
+            j += 1
+        a, b, c = tri
+        g.add_edges_from([(a, b), (b, c), (a, c)])
+        tread_triangles.append(tri)
+    if (i, j) != (m, L):
+        return None
+    return g, outer, tread_triangles
+
+
+def cap_and_validate(g, outer):
+    """Cap the outer cycle with apex N; require a simple closed triangulation."""
+    h = g.copy()
+    for t in range(len(outer)):
+        h.add_edges_from([("N", outer[t]), ("N", outer[(t + 1) % len(outer)])])
+    if not nx.check_planarity(h)[0]:
+        return None
+    V, E = h.number_of_nodes(), h.number_of_edges()
+    if E != 3 * V - 6:          # maximal planar == triangulation
+        return None
+    return h
+
+
+def find_construction(m: int):
+    L = len(INNER_WALK)
+    for combo in itertools.combinations(range(m + L), L):
+        path = "".join("I" if t in combo else "O" for t in range(m + L))
+        built = build_tread(m, path)
+        if built is None:
+            continue
+        g, outer, tris = built
+        h = cap_and_validate(g, outer)
+        if h is not None:
+            return h, outer, tris, path
+    return None
+
+
+def run():
+    for m in (4, 5, 6, 7):
+        found = find_construction(m)
+        if found:
+            break
+    if not found:
+        print("no valid bite-with-singletons triangulation found")
+        return 1
+    h, outer, tris, path = found
+    M = medial_graph(h)
+
+    annular = set()
+    for tri in tris:
+        a, b, c = tri
+        for e in (ekey(a, b), ekey(b, c), ekey(a, c)):
+            x, y = e
+            xo, yo = x in outer, y in outer
+            if (xo and not yo) or (yo and not xo):
+                annular.add(e)
+
+    n_bite_nbrs = sum(1 for nb in M.neighbors(BITE_EDGE) if nb in annular)
+    up = [ekey(outer[t], outer[(t + 1) % len(outer)]) for t in range(len(outer))]
+    up = [e for e in up if e in M]
+
+    print(f"m={len(outer)} path={path}  |V(G)|={h.number_of_nodes()} "
+          f"|M(G)|={M.number_of_nodes()}")
+    print(f"bite edge {BITE_EDGE}: annular neighbours={n_bite_nbrs} (4 => bite)")
+    print(f"up apexes={len(up)}  singleton down apexes={SINGLETON_DOWN}")
+
+    colorings = proper_3_colorings(M, limit=50000)
+    bad_outer = bad_bitegap = 0
+    for col in colorings:
+        if any(sum(1 for e in up if col[e] in p) % 2 for p in PAIRS):
+            bad_outer += 1
+        if any(sum(1 for e in SINGLETON_DOWN if col[e] in p) % 2 for p in PAIRS):
+            bad_bitegap += 1
+
+    print(f"colourings tested={len(colorings)}")
+    print(f"    outer face unbalanced: {bad_outer}")
+    print(f"    bite-gap face (3 singletons) unbalanced: {bad_bitegap}")
+    print()
+    ok = (bad_outer == 0 and bad_bitegap == 0)
+    print("Remark 5.8 holds on this bite-with-singletons tread"
+          if ok else "Remark 5.8 FAILS on this bite-with-singletons tread")
+    return 0 if ok else 1
+
+
+if __name__ == "__main__":
+    run()

papers/medial_tire_decompositions_of_plane_triangulations/paper.aux

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papers/medial_tire_decompositions_of_plane_triangulations/paper.tex

@@ -841,12 +841,14 @@ meets each level cycle in an even number of $P$-coloured incidences; for
 a given valid face these incidences are exactly its incident tooth
 apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
 
-This argument is verified computationally for bite-free pieces: across
+This argument is verified computationally.  For bite-free pieces---capped
-all proper $3$-colourings of the capped triangulated annuli on annular
+triangulated annuli on annular cycles of length $6,8,10,12$---every proper
-cycles of length $6,8,10,12$, every restriction to the tire piece is
+$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring.  The same
-Kempe-balanced.  The case with bites, where the inner level cycle splits
+holds for pieces carrying a bite, including the case where singleton down
-into several child level cycles, is consistent with the same mechanism
+teeth lie in the bite's inner-gap face: there the inner level cycle splits
-but is not yet checked end to end.
+into a child level cycle per gap, and conservation across each child cycle
+supplies the parity (in the checked example the three singleton down apexes
+of a bite gap are a rainbow in every restriction).
 \end{remark}
 
 More generally, let $T$ be a medial tire region with boundary