dual_decomposition: swap algorithm trace to n=14 + final-graph conjecture

- Replace the dodecahedron trace at the end of section 3 with the n=14
  triangulation found by search_kempe_property.py: its H_1 admits a
  proper 3-edge-colouring satisfying both chord-apex and Kempe-cycle
  conditions (Lemmas 2.6, 2.7).
- experiments/draw_iterated_reduction_n14.py: rebuilds fig_alg_step{0,1,2}
  with Tutte barycentric layouts (outer face chosen to keep v_n in the
  interior); also runs the algorithm to completion, checking chord-apex +
  Kempe at each step (step 1 satisfies all; step 2 fails chord-apex;
  step 3 terminates).
- Add Conjecture 3.4: G is a minimal counterexample iff no proper
  3-edge-colouring of the final reduced graph H_{t*} has all (spike_t,
  merged_t) pairs in distinct colours.

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

papers/dual_decomposition_minimal_counterexamples/experiments/draw_iterated_reduction_n14.py

@@ -0,0 +1,520 @@
+"""Draw the iterated reduction trace on the smallest triangulation where
+the chord-apex + Kempe-cycle property is satisfied: the first min-degree-5
+plantri triangulation on n = 14 vertices, found by search_kempe_property.py.
+
+Overwrites fig_alg_step{0,1,2}.png in the paper directory with this
+triangulation's trace (replacing the dodecahedron version).
+
+Run with: sage experiments/draw_iterated_reduction_n14.py
+"""
+from sage.all import Graph
+from sage.graphs.graph_generators import graphs
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+import math
+import os
+
+
+def tutte_layout(G_sage, avoid_verts=None, iterations=300):
+    """Tutte's barycentric embedding: pick the largest face whose vertex set
+    avoids `avoid_verts` as the outer face, place its vertices on a regular
+    polygon, then iterate each interior vertex to the barycenter of its
+    neighbors. For 3-connected planar graphs this converges to the unique
+    straight-line planar embedding with the chosen outer face --- balanced
+    by construction and free of edge crossings."""
+    avoid = set(avoid_verts or ())
+    candidates = []
+    for face in G_sage.faces():
+        verts = [u for (u, v) in face]
+        if not (set(verts) & avoid):
+            candidates.append(verts)
+    if not candidates:
+        outer = [u for (u, v) in max(G_sage.faces(), key=len)]
+    else:
+        outer = max(candidates, key=len)
+    n_outer = len(outer)
+    pos = {}
+    for k, v in enumerate(outer):
+        ang = 2 * math.pi * k / n_outer + math.pi / 2
+        pos[v] = (math.cos(ang), math.sin(ang))
+    interior = [v for v in G_sage.vertex_iterator() if v not in pos]
+    for v in interior:
+        pos[v] = (0.0, 0.0)
+    for _ in range(iterations):
+        new_pos = dict(pos)
+        for v in interior:
+            nbrs = list(G_sage.neighbor_iterator(v))
+            sx = sum(pos[w][0] for w in nbrs) / len(nbrs)
+            sy = sum(pos[w][1] for w in nbrs) / len(nbrs)
+            new_pos[v] = (sx, sy)
+        pos = new_pos
+    return pos
+
+OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
+
+C = ['#dc2626', '#16a34a', '#2563eb']
+GRAY = '#9ca3af'
+DARK = '#374151'
+HIGHLIGHT = '#fef3c7'
+SHADE = '#fef3c7'
+
+
+def dual_of(G):
+    faces = G.faces()
+    edge_to_faces = {}
+    for fi, face in enumerate(faces):
+        for u, v in face:
+            e = frozenset((u, v))
+            edge_to_faces.setdefault(e, []).append(fi)
+    dual_edges = []
+    for e, fs in edge_to_faces.items():
+        if len(fs) == 2:
+            dual_edges.append((fs[0], fs[1]))
+    return Graph(dual_edges, multiedges=False, loops=False)
+
+
+def apply_reduction(G, face, i, v_n_label):
+    boundary = [u for (u, v) in face]
+    if len(set(boundary)) != 5:
+        return None
+    A = []
+    for B_k in boundary:
+        outer = [w for w in G.neighbor_iterator(B_k) if w not in boundary]
+        if len(outer) != 1:
+            return None
+        A.append(outer[0])
+    if len(set(A)) != 5:
+        return None
+    if A[(i + 3) % 5] == A[(i + 4) % 5]:
+        return None
+    H = G.copy()
+    for v in boundary:
+        H.delete_vertex(v)
+    H.add_vertex(v_n_label)
+    side_0 = (v_n_label, A[i % 5])
+    spike  = (v_n_label, A[(i + 1) % 5])
+    side_1 = (v_n_label, A[(i + 2) % 5])
+    merged = (A[(i + 3) % 5], A[(i + 4) % 5])
+    H.add_edges([side_0, spike, side_1, merged])
+    if H.has_multiple_edges() or H.has_loops():
+        return None
+    if not H.is_planar(set_embedding=True):
+        return None
+    if not all(H.degree(v) == 3 for v in H.vertex_iterator()):
+        return None
+    named = {
+        'spike':  frozenset(spike),
+        'side_0': frozenset(side_0),
+        'side_1': frozenset(side_1),
+        'merged': frozenset(merged),
+    }
+    return H, named, boundary, A
+
+
+def proper_3_edge_colorings(G):
+    edges = list(G.edges(labels=False))
+    n_edges = len(edges)
+    adj = [[] for _ in range(n_edges)]
+    for i in range(n_edges):
+        u, v = edges[i][0], edges[i][1]
+        for j in range(i):
+            x, y = edges[j][0], edges[j][1]
+            if u in (x, y) or v in (x, y):
+                adj[i].append(j)
+                adj[j].append(i)
+    coloring = [-1] * n_edges
+
+    def back(k):
+        if k == n_edges:
+            yield tuple(coloring)
+            return
+        for c in range(3):
+            if all(coloring[j] != c for j in adj[k]):
+                coloring[k] = c
+                yield from back(k + 1)
+        coloring[k] = -1
+
+    return edges, back(0)
+
+
+def kempe_cycle(edges, coloring, start_idx, color_pair):
+    a, b = color_pair
+    in_sub = [i for i in range(len(edges)) if coloring[i] in (a, b)]
+    if start_idx not in in_sub:
+        return None
+    visited = {start_idx}
+    stack = [start_idx]
+    while stack:
+        cur = stack.pop()
+        u, v = edges[cur][0], edges[cur][1]
+        for j in in_sub:
+            if j in visited:
+                continue
+            x, y = edges[j][0], edges[j][1]
+            if u in (x, y) or v in (x, y):
+                visited.add(j)
+                stack.append(j)
+    return visited
+
+
+def matches_property(edges, col, named):
+    idx = {}
+    for ii, e in enumerate(edges):
+        es = frozenset((e[0], e[1]))
+        for role, ns in named.items():
+            if es == ns:
+                idx[role] = ii
+    if len(idx) != 4:
+        return False
+    c_spike = col[idx['spike']]
+    c_merged = col[idx['merged']]
+    if c_spike != c_merged:
+        return False
+    c_s0 = col[idx['side_0']]
+    c_s1 = col[idx['side_1']]
+    kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
+    if idx['side_0'] not in kc0 or idx['merged'] not in kc0:
+        return False
+    kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
+    if idx['side_1'] not in kc1 or idx['merged'] not in kc1:
+        return False
+    return True
+
+
+def find_first_match():
+    """Iterate over (G, face, i_red, coloring) and return the first hit."""
+    for G in graphs.triangulations(14, minimum_degree=5):
+        if not G.is_planar(set_embedding=True):
+            continue
+        D = dual_of(G)
+        D.is_planar(set_embedding=True)
+        for face in D.faces():
+            if len(face) != 5:
+                continue
+            for i_red in range(5):
+                res = apply_reduction(D, face, i_red, '__v_n_1__')
+                if res is None:
+                    continue
+                H, named, boundary, A = res
+                edges, gen = proper_3_edge_colorings(H)
+                for col in gen:
+                    if matches_property(edges, col, named):
+                        coloring_dict = {frozenset((e[0], e[1])): c
+                                         for e, c in zip(edges, col)}
+                        return G, D, face, i_red, H, named, boundary, A, coloring_dict
+    return None
+
+
+def draw_graph(ax, G, pos, *, coloring=None, protected=None,
+               shade_vertices=None, vn_labels=None):
+    if shade_vertices:
+        poly = [pos[v] for v in shade_vertices]
+        ax.add_patch(Polygon(poly, closed=True, facecolor=SHADE,
+                             edgecolor='none', zorder=0))
+    protected = protected or set()
+    vn_labels = vn_labels or {}
+    for u, v, _ in G.edges():
+        e = frozenset([u, v])
+        c = C[coloring[e]] if (coloring is not None and e in coloring) else GRAY
+        lw = 3.8 if e in protected else 1.4
+        (x0, y0), (x1, y1) = pos[u], pos[v]
+        ax.plot([x0, x1], [y0, y1], color=c, lw=lw, zorder=2)
+    for v in G.vertices(sort=False):
+        x, y = pos[v]
+        if v in vn_labels:
+            ax.scatter(x, y, s=320, color=HIGHLIGHT, marker='s',
+                       edgecolors='black', linewidths=1.2, zorder=4)
+            ax.annotate(vn_labels[v], (x, y),
+                        textcoords='offset points', xytext=(16, 16),
+                        ha='left', fontsize=14, fontweight='bold',
+                        color=DARK, zorder=6,
+                        bbox=dict(boxstyle='round,pad=0.2', fc='white',
+                                  ec=DARK, lw=0.6))
+        else:
+            ax.scatter(x, y, s=60, color=DARK, zorder=3)
+    ax.set_aspect('equal')
+    ax.axis('off')
+
+
+def main():
+    print("Searching for the first match at n = 14 ...")
+    result = find_first_match()
+    if result is None:
+        print("No match found at n = 14.")
+        return
+    G14, D, face, i_red, H1, named1, boundary1, A1, coloring1 = result
+    print(f"Found at i_red = {i_red}")
+    print(f"  G (n=14):  |V|={G14.order()}, |E|={G14.size()}, "
+          f"min_deg={min(G14.degree())}")
+    print(f"  D = G':    |V|={D.order()}, |E|={D.size()}")
+    print(f"  H_1:       |V|={H1.order()}, |E|={H1.size()}")
+
+    # Relabel H_1 in place so all vertex labels are comparable integers
+    # (Sage's planar layout and face enumeration need comparable labels).
+    # Translate coloring1 and named1 accordingly.
+    H1_relabel_map = {v: i for i, v in enumerate(H1.vertex_iterator())}
+    H1.relabel(perm=H1_relabel_map, inplace=True)
+    vn1_int = H1_relabel_map['__v_n_1__']
+    coloring1 = {frozenset(H1_relabel_map[u] for u in e): c
+                 for e, c in coloring1.items()}
+    named1 = {role: frozenset(H1_relabel_map[u] for u in e)
+              for role, e in named1.items()}
+
+    D.is_planar(set_embedding=True)
+    D_layout = tutte_layout(D, avoid_verts=set(u for (u, v) in face))
+    H1.is_planar(set_embedding=True)
+    H1_layout = tutte_layout(H1, avoid_verts={vn1_int})
+
+    boundary_face_verts = [u for (u, v) in face]
+    fig, ax = plt.subplots(figsize=(8, 8))
+    draw_graph(ax, D, D_layout, shade_vertices=boundary_face_verts)
+    fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step0.png'),
+                dpi=170, bbox_inches='tight')
+    plt.close(fig)
+    print("Wrote fig_alg_step0.png")
+
+    E = set(named1.values())
+    fig, ax = plt.subplots(figsize=(8, 8))
+    draw_graph(ax, H1, H1_layout, coloring=coloring1, protected=E,
+               vn_labels={vn1_int: '$v_n^{(1)}$'})
+    fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step1.png'),
+                dpi=170, bbox_inches='tight')
+    plt.close(fig)
+    print("Wrote fig_alg_step1.png")
+
+    # ----- Step 2: try to continue -----
+    H1.is_planar(set_embedding=True)
+    chosen2 = None
+    for face2 in H1.faces():
+        if len(face2) != 5:
+            continue
+        boundary2 = [u for (u, v) in face2]
+        boundary2_edges = [frozenset([u, v]) for (u, v) in face2]
+        externals2 = []
+        A2 = []
+        valid_face = True
+        for B_k in boundary2:
+            outer = [w for w in H1.neighbor_iterator(B_k) if w not in boundary2]
+            if len(outer) != 1:
+                valid_face = False
+                break
+            externals2.append(frozenset([B_k, outer[0]]))
+            A2.append(outer[0])
+        if not valid_face:
+            continue
+        if any(e in E for e in boundary2_edges + externals2):
+            continue
+        # find valid i_t
+        f_vec = [coloring1[e] for e in externals2]
+        for i_t in range(5):
+            if f_vec[(i_t + 3) % 5] != f_vec[(i_t + 4) % 5]:
+                continue
+            if len({f_vec[i_t], f_vec[(i_t + 1) % 5], f_vec[(i_t + 2) % 5]}) != 3:
+                continue
+            if A2[(i_t + 3) % 5] == A2[(i_t + 4) % 5]:
+                continue
+            chosen2 = (face2, i_t, boundary2, externals2, A2)
+            break
+        if chosen2 is not None:
+            break
+
+    if chosen2 is None:
+        # algorithm terminates at H_1
+        fig, ax = plt.subplots(figsize=(8, 8))
+        ax.text(0.5, 0.5,
+                "Algorithm terminates at $H_1$:\n"
+                "no pentagonal face of $H_1$ has all\n"
+                "ten incident edges outside $E$.",
+                ha='center', va='center', fontsize=18, color=DARK,
+                transform=ax.transAxes,
+                bbox=dict(boxstyle='round,pad=0.6', fc=HIGHLIGHT,
+                          ec=DARK, lw=1.0))
+        ax.set_aspect('equal')
+        ax.axis('off')
+        fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
+                    dpi=170, bbox_inches='tight')
+        plt.close(fig)
+        print("Wrote fig_alg_step2.png (termination card)")
+        print("  Algorithm terminates at H_1: no safe pentagonal face.")
+        return
+
+    face2, i_t, boundary2, externals2, A2 = chosen2
+    print(f"Step 2: safe face found, i_t = {i_t}")
+
+    H2 = H1.copy()
+    for v in boundary2:
+        H2.delete_vertex(v)
+    # use a fresh int label for v_n^(2)
+    v_n_2 = max(H1.vertices(sort=False)) + 1
+    H2.add_vertex(v_n_2)
+    side_0_2 = (v_n_2, A2[i_t])
+    spike_2  = (v_n_2, A2[(i_t + 1) % 5])
+    side_1_2 = (v_n_2, A2[(i_t + 2) % 5])
+    merged_2 = (A2[(i_t + 3) % 5], A2[(i_t + 4) % 5])
+    H2.add_edges([side_0_2, spike_2, side_1_2, merged_2])
+    H2.is_planar(set_embedding=True)
+
+    coloring2 = {e: c for e, c in coloring1.items()
+                 if not any(u in boundary2 for u in e)}
+    coloring2[frozenset(side_0_2)] = coloring1[externals2[i_t]]
+    coloring2[frozenset(spike_2)]  = coloring1[externals2[(i_t + 1) % 5]]
+    coloring2[frozenset(side_1_2)] = coloring1[externals2[(i_t + 2) % 5]]
+    coloring2[frozenset(merged_2)] = coloring1[externals2[(i_t + 3) % 5]]
+
+    E |= {frozenset(side_0_2), frozenset(spike_2),
+          frozenset(side_1_2), frozenset(merged_2)}
+
+    H2_layout = tutte_layout(H2, avoid_verts={vn1_int, v_n_2})
+    fig, ax = plt.subplots(figsize=(8, 8))
+    draw_graph(ax, H2, H2_layout, coloring=coloring2, protected=E,
+               vn_labels={vn1_int: '$v_n^{(1)}$',
+                          v_n_2: '$v_n^{(2)}$'})
+    fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
+                dpi=170, bbox_inches='tight')
+    plt.close(fig)
+    print(f"Wrote fig_alg_step2.png: H_2 with |V|={H2.order()}, "
+          f"|E|={H2.size()}, |protected|={len(E)}")
+
+    # --- continue running to completion, checking Kempe condition each step --
+    print()
+    print("=" * 72)
+    print("Running algorithm to completion, checking chord-apex + Kempe at "
+          "each step.")
+    print("=" * 72)
+    # Step 1 status (by construction this is the matching coloring)
+    cond1 = check_step_conditions(H1, coloring1, named1)
+    print(f"  step t = 1: |V|={H1.order():>3}, |E_graph|={H1.size():>3}, "
+          f"|E_prot|=  4 (initial)"
+          f" | chord-apex: {cond1['chord_apex']}, "
+          f"side_0-Kempe: {cond1['kc_side_0']}, "
+          f"side_1-Kempe: {cond1['kc_side_1']}")
+    run_to_completion_from(H2, coloring2, E,
+                           {'spike': frozenset(spike_2),
+                            'side_0': frozenset(side_0_2),
+                            'side_1': frozenset(side_1_2),
+                            'merged': frozenset(merged_2)},
+                           start_t=2)
+
+
+def check_step_conditions(H, coloring, named):
+    """Given an H_t and the *just-added* spike/side_0/side_1/merged, check
+    whether chord-apex and the two Kempe-cycle conditions hold."""
+    edges = list(H.edges(labels=False))
+    edges_fs = [frozenset((u, v)) for (u, v) in edges]
+    col = [coloring[e] for e in edges_fs]
+    idx = {role: edges_fs.index(e) for role, e in named.items()}
+    c_spike  = col[idx['spike']]
+    c_merged = col[idx['merged']]
+    chord_apex = (c_spike == c_merged)
+    if not chord_apex:
+        return {'chord_apex': False, 'kc_side_0': False, 'kc_side_1': False}
+    c_s0 = col[idx['side_0']]
+    c_s1 = col[idx['side_1']]
+    kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
+    kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
+    kc_side_0 = (idx['side_0'] in kc0 and idx['merged'] in kc0)
+    kc_side_1 = (idx['side_1'] in kc1 and idx['merged'] in kc1)
+    return {'chord_apex': True, 'kc_side_0': kc_side_0, 'kc_side_1': kc_side_1}
+
+
+def find_safe_face(H, protected):
+    """Return (face, externals, A) for some safe pentagonal face avoiding
+    `protected`, or None."""
+    for face in H.faces():
+        if len(face) != 5:
+            continue
+        boundary = [u for (u, v) in face]
+        boundary_edges = [frozenset([u, v]) for (u, v) in face]
+        externals = []
+        A = []
+        valid = True
+        for B_k in boundary:
+            outer = [w for w in H.neighbor_iterator(B_k) if w not in boundary]
+            if len(outer) != 1:
+                valid = False
+                break
+            externals.append(frozenset([B_k, outer[0]]))
+            A.append(outer[0])
+        if not valid:
+            continue
+        if any(e in protected for e in boundary_edges + externals):
+            continue
+        return face, boundary, externals, A
+    return None
+
+
+def run_to_completion_from(H, coloring, E, last_named, start_t):
+    """Continue iterating from H_{start_t}. The 'last_named' dict carries
+    the spike/side/merged of step `start_t` so we can report its Kempe
+    status. Print a row per step."""
+    t = start_t
+    print(f"  step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
+          f"|E_prot|={len(E):>3}", end='')
+    cond = check_step_conditions(H, coloring, last_named)
+    print(f"  | chord-apex: {cond['chord_apex']}, "
+          f"side_0-Kempe: {cond['kc_side_0']}, "
+          f"side_1-Kempe: {cond['kc_side_1']}")
+
+    while True:
+        H.is_planar(set_embedding=True)
+        res = find_safe_face(H, E)
+        if res is None:
+            print(f"  step t = {t + 1}: no safe pentagonal face --> "
+                  f"algorithm terminates at H_{t}.")
+            return
+        face, boundary, externals, A = res
+        f_vec = [coloring[e] for e in externals]
+        i_t = None
+        for i in range(5):
+            if f_vec[(i + 3) % 5] != f_vec[(i + 4) % 5]:
+                continue
+            if len({f_vec[i], f_vec[(i + 1) % 5], f_vec[(i + 2) % 5]}) != 3:
+                continue
+            if A[(i + 3) % 5] == A[(i + 4) % 5]:
+                continue
+            i_t = i
+            break
+        if i_t is None:
+            print(f"  step t = {t + 1}: f = {f_vec}, no valid index --> "
+                  f"terminate (Lemma 2.4 violation? Probably a parallel-edge "
+                  f"or other degenerate case).")
+            return
+        t += 1
+        v_n_new = max(H.vertices(sort=False)) + 1 if all(
+            isinstance(v, int) for v in H.vertex_iterator()) else f'vn{t}'
+        H_new = H.copy()
+        for v in boundary:
+            H_new.delete_vertex(v)
+        H_new.add_vertex(v_n_new)
+        side_0 = (v_n_new, A[i_t])
+        spike  = (v_n_new, A[(i_t + 1) % 5])
+        side_1 = (v_n_new, A[(i_t + 2) % 5])
+        merged = (A[(i_t + 3) % 5], A[(i_t + 4) % 5])
+        H_new.add_edges([side_0, spike, side_1, merged])
+        H = H_new
+        coloring = {e: c for e, c in coloring.items()
+                    if not any(u in boundary for u in e)}
+        coloring[frozenset(side_0)] = coloring[externals[i_t]] \
+            if frozenset(externals[i_t]) in coloring else f_vec[i_t]
+        # safer: directly use f_vec
+        coloring[frozenset(side_0)] = f_vec[i_t]
+        coloring[frozenset(spike)]  = f_vec[(i_t + 1) % 5]
+        coloring[frozenset(side_1)] = f_vec[(i_t + 2) % 5]
+        coloring[frozenset(merged)] = f_vec[(i_t + 3) % 5]
+        named = {
+            'spike':  frozenset(spike),
+            'side_0': frozenset(side_0),
+            'side_1': frozenset(side_1),
+            'merged': frozenset(merged),
+        }
+        E |= set(named.values())
+        cond = check_step_conditions(H, coloring, named)
+        print(f"  step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
+              f"|E_prot|={len(E):>3}, i_t = {i_t}", end='')
+        print(f"  | chord-apex: {cond['chord_apex']}, "
+              f"side_0-Kempe: {cond['kc_side_0']}, "
+              f"side_1-Kempe: {cond['kc_side_1']}")
+
+
+if __name__ == '__main__':
+    main()

papers/dual_decomposition_minimal_counterexamples/paper.aux

@@ -22,6 +22,7 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G' = $ dodecahedron (dual of the icosahedron). \emph  {Left:} $G'$ (20 vertices, 30 edges), with $F_v$ (the inner pentagon) shaded as the face chosen for the first reduction. \emph  {Centre:} $H_1$ (16 vertices, 24 edges) after step\nonbreakingspace (1) with $i_1 = 0$, $3$-edge-coloured by Sage; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker. \emph  {Right:} $H_2$ (12 vertices, 18 edges) after step\nonbreakingspace (3) with $i_t = 0$; the only safe pentagonal face in $H_1$ was the outer pentagon, whose deletion produces $v_n^{(2)}$ and a second chord, giving eight protected edges. No safe pentagonal face remains, so the algorithm terminates. The generating script is \texttt  {experiments/draw\_iterated\_reduction.py}.}}{8}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G'=\mathrm  {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex condition (Lemma\nonbreakingspace 2.6\hbox {}) and the Kempe-cycle condition (Lemma\nonbreakingspace 2.7\hbox {}), found by \texttt  {experiments/search\_kempe\_property.py}. \emph  {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph  {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph  {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt  {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{8}{}\protected@file@percent }
 \newlabel{fig:iterated-reduction-trace}{{3}{8}}
+\newlabel{conj:no-all-distinct-coloring}{{3.4}{8}}
 \gdef \@abspage@last{8}

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papers/dual_decomposition_minimal_counterexamples/paper.tex

@@ -12,6 +12,7 @@
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
 \newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{conjecture}[theorem]{Conjecture}
 
 \theoremstyle{definition}
 \newtheorem{definition}[theorem]{Definition}
@@ -478,17 +479,36 @@ any further structure to $\varphi_t$ for $t \geq 2$ is left open.
 \includegraphics[width=0.32\textwidth]{fig_alg_step0.png}\hfill
 \includegraphics[width=0.32\textwidth]{fig_alg_step1.png}\hfill
 \includegraphics[width=0.32\textwidth]{fig_alg_step2.png}
-\caption{Algorithm~\ref{alg:iterated-reduction} on $G' = $ dodecahedron
+\caption{Algorithm~\ref{alg:iterated-reduction} on $G'=\mathrm{dual}(G)$, where
-(dual of the icosahedron). \emph{Left:} $G'$ (20 vertices, 30 edges), with
+$G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and
-$F_v$ (the inner pentagon) shaded as the face chosen for the first reduction.
+$\varphi_1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies
-\emph{Centre:} $H_1$ (16 vertices, 24 edges) after step~(1) with $i_1 = 0$,
+both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and the Kempe-cycle
-$3$-edge-coloured by Sage; the four edges around $v_n^{(1)}$ in $E$ are drawn
+condition (Lemma~\ref{lem:kempe-spike}), found by
-thicker. \emph{Right:} $H_2$ (12 vertices, 18 edges) after step~(3) with
+\texttt{experiments/search\_kempe\_property.py}. \emph{Left:} $G'$
-$i_t = 0$; the only safe pentagonal face in $H_1$ was the outer pentagon,
+($24$ vertices, $36$ edges) with the chosen pentagonal face shaded.
-whose deletion produces $v_n^{(2)}$ and a second chord, giving eight protected
+\emph{Centre:} $H_1$ ($20$ vertices, $30$ edges) after step~(1) with
-edges. No safe pentagonal face remains, so the algorithm terminates. The
+$i_1 = 1$, $3$-edge-coloured by $\varphi_1$; the four edges around
-generating script is \texttt{experiments/draw\_iterated\_reduction.py}.}
+$v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share
+the colour green. \emph{Right:} $H_2$ ($16$ vertices, $24$ edges) after
+step~(3) with $i_t = 3$; eight edges are protected, and the algorithm
+terminates one step later (no remaining safe pentagonal face in $H_2$).
+The generating script is
+\texttt{experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte
+barycentric embeddings with the outer face picked to keep $v_n^{(1)},
+v_n^{(2)}$ in the interior.}
 \label{fig:iterated-reduction-trace}
 \end{figure}
 
+\begin{conjecture}
+\label{conj:no-all-distinct-coloring}
+Let $G$ be a maximal planar graph of minimum degree $\geq 5$, and let
+$H_{t^*}$ be the final reduced graph produced by some terminating execution
+of Algorithm~\ref{alg:iterated-reduction} on $G$, with the corresponding
+sequence of named pairs $(\mathrm{spike}_t, \mathrm{merged}_t)$ for
+$t = 1, \dots, t^*$. Then $G$ is a minimal counterexample to the Four Colour
+Theorem if and only if there is no proper $3$-edge-colouring of $H_{t^*}$
+under which $\mathrm{spike}_t$ and $\mathrm{merged}_t$ receive different
+colours for every $t \in \{1, \dots, t^*\}$.
+\end{conjecture}
+
 \end{document}