math-research/papers/face_monochromatic_pairs/experiments/check_conj_face_kempe.py

"""Test Conjecture 3.6: for every reduced dual of a minimal counterexample,
every proper 3-edge-coloring has a face with two same-color edges that share
a Kempe cycle with the merged edge.

We can't run on actual counterexamples (none exist by the 4CT). Instead we
test on the structural surrogates: proper 3-edge-colorings of reduced duals
that *satisfy* chord-apex (Lemma 2.6) and the Kempe-cycle condition
(Lemma 2.7). These are the kinds of colorings a counterexample's reduced
dual would be forced to admit.

For each such (G, face, i_red, phi), check whether some face F of H_1 and
some pair of edges (e1, e2) in partial F satisfy:
  - phi(e1) = phi(e2)
  - e1, e2, and merged all lie on a common {a,b}-Kempe cycle.

If conjecture FAILS for some coloring, we have empirical evidence against.
If every coloring passes, evidence in favour.

Run with: sage experiments/check_conj_face_kempe.py
"""
from sage.all import Graph
from sage.graphs.graph_generators import graphs


def dual_of(G):
    G.is_planar(set_embedding=True)
    faces = G.faces()
    edge_to_faces = {}
    for fi, face in enumerate(faces):
        for u, v in face:
            edge_to_faces.setdefault(frozenset((u, v)), []).append(fi)
    return Graph(
        [(fs[0], fs[1]) for fs in edge_to_faces.values() if len(fs) == 2],
        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 or 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])
    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
    return {
        'H': H,
        'named': {
            'spike':  frozenset(spike), 'side_0': frozenset(side_0),
            'side_1': frozenset(side_1), 'merged': frozenset(merged),
        },
    }


def proper_3_edge_colorings(G):
    edges = list(G.edges(labels=False))
    n = len(edges)
    adj = [[] for _ in range(n)]
    for i in range(n):
        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
    results = []

    def back(k):
        if k == n:
            results.append(tuple(coloring))
            return
        for c in range(3):
            if all(coloring[j] != c for j in adj[k]):
                coloring[k] = c
                back(k + 1)
        coloring[k] = -1
    back(0)
    return edges, results


def kempe_cycle(edges, coloring, start_idx, color_pair):
    a, b = color_pair
    if coloring[start_idx] not in (a, b):
        return set()
    in_sub = set(i for i in range(len(edges)) if coloring[i] in (a, b))
    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 edge_idx(edges, e_frozen):
    for i, e in enumerate(edges):
        if frozenset((e[0], e[1])) == e_frozen:
            return i
    return None


def matches_chord_apex_kempe(edges, col, named):
    idx = {role: edge_idx(edges, ns) for role, ns in named.items()}
    if any(v is None for v in idx.values()): 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 conjecture_holds_for(H, edges, col, named):
    """Returns (F, e1, e2, kc) if some face F of H has two same-colour edges
    (e1, e2), neither equal to merged, both on a Kempe cycle kc through
    merged, AND exactly one edge of partial F lies between e1 and e2 along
    one of the two arcs of partial F. Else None."""
    merged_idx = edge_idx(edges, named['merged'])
    c_merged = col[merged_idx]
    kempe_cycles = []
    for c_prime in range(3):
        if c_prime == c_merged: continue
        kc = kempe_cycle(edges, col, merged_idx, (c_merged, c_prime))
        kempe_cycles.append(kc)
    for face in H.faces():
        face_edge_indices = []
        for u, v in face:
            ei = edge_idx(edges, frozenset((u, v)))
            if ei is not None:
                face_edge_indices.append(ei)
        n_face = len(face_edge_indices)
        for i in range(n_face):
            for j in range(i + 1, n_face):
                e1, e2 = face_edge_indices[i], face_edge_indices[j]
                if e1 == merged_idx or e2 == merged_idx: continue
                if col[e1] != col[e2]: continue
                # exactly one edge between e1 and e2 in one arc
                gap_a = (j - i - 1)
                gap_b = (n_face - 2 - gap_a)
                if gap_a != 1 and gap_b != 1:
                    continue
                for kc in kempe_cycles:
                    if e1 in kc and e2 in kc:
                        return face, e1, e2, kc
    return None


def main():
    for n in [12, 14, 15]:
        print(f"=== n = {n} ===")
        try:
            triangulations = list(graphs.triangulations(n, minimum_degree=5))
        except Exception as ex:
            print(f"  cannot enumerate: {ex}"); continue
        for tri_idx, G in enumerate(triangulations, start=1):
            G.is_planar(set_embedding=True)
            D = dual_of(G); D.is_planar(set_embedding=True)
            for face_no, face in enumerate(
                    f for f in D.faces() if len(f) == 5):
                for i_red in range(5):
                    res = apply_reduction(D, face, i_red, 9999)
                    if res is None: continue
                    H = res['H']; named = res['named']
                    H.is_planar(set_embedding=True)
                    edges, colorings = proper_3_edge_colorings(H)
                    cand = [c for c in colorings
                            if matches_chord_apex_kempe(edges, c, named)]
                    if not cand: continue
                    n_pass = sum(1 for c in cand
                                 if conjecture_holds_for(H, edges, c, named)
                                    is not None)
                    n_fail = len(cand) - n_pass
                    status = "PASSED" if n_fail == 0 else "*** FAILED ***"
                    print(f"  n={n} tri#{tri_idx} face#{face_no} i_red={i_red}: "
                          f"{len(cand)} chord-apex+Kempe colorings; "
                          f"{n_pass} satisfy conjecture, {n_fail} fail. "
                          f"{status}")


if __name__ == '__main__':
    main()