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

"""Probe: even when phi_t* is in a Kempe class with no all-distinct, does
H_t* itself admit an all-distinct coloring in SOME (other) Kempe class?

Reuses the Conj 3.6 counterexample: n=14, tri#1, i_red=0, phi_1 from the
chord-apex+Kempe colorings. Enumerates every proper 3-edge-coloring of
H_t*, marks the Kempe classes, and for each class reports whether it
contains an all-distinct coloring.

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


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)
    dual_edges = [(fs[0], fs[1]) for fs in edge_to_faces.values() if len(fs) == 2]
    return Graph(dual_edges, multiedges=False, loops=False)


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 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])
    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, 'A': A,
        'named': {
            'spike':  frozenset(spike),
            'side_0': frozenset(side_0),
            'side_1': frozenset(side_1),
            'merged': frozenset(merged),
        },
    }


def find_safe_pentagonal_face(G, protected):
    for face in G.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 = []
        ok = True
        for B_k in boundary:
            outer = [w for w in G.neighbor_iterator(B_k) if w not in boundary]
            if len(outer) != 1:
                ok = False
                break
            externals.append(frozenset([B_k, outer[0]]))
            A.append(outer[0])
        if not ok:
            continue
        if any(e in protected for e in boundary_edges + externals):
            continue
        return boundary, externals, A
    return None


def valid_index(f_vec, A):
    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
        return i
    return None


def run_algorithm_to_completion(H, phi_1_dict, named_step1, vn_start=10000):
    H = H.copy()
    coloring = dict(phi_1_dict)
    all_named = [named_step1]
    E = set(named_step1.values())
    next_vn = vn_start
    while True:
        H.is_planar(set_embedding=True)
        result = find_safe_pentagonal_face(H, E)
        if result is None:
            break
        boundary, externals, A = result
        f_vec = [coloring[e] for e in externals]
        i_t = valid_index(f_vec, A)
        if i_t is None:
            break
        v_n = next_vn
        next_vn += 1
        H_new = H.copy()
        for v in boundary:
            H_new.delete_vertex(v)
        H_new.add_vertex(v_n)
        side_0 = (v_n, A[i_t])
        spike  = (v_n, A[(i_t + 1) % 5])
        side_1 = (v_n, 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])
        if H_new.has_multiple_edges() or H_new.has_loops():
            break
        if not H_new.is_planar(set_embedding=True):
            break
        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)] = 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())
        all_named.append(named)
    return H, coloring, all_named


def is_all_distinct(edges, col, all_named):
    for named in all_named:
        i_s = edge_idx(edges, named['spike'])
        i_m = edge_idx(edges, named['merged'])
        if i_s is None or i_m is None:
            return False
        if col[i_s] == col[i_m]:
            return False
    return True


def main():
    print("Reconstructing Conj 3.6 counterexample: n=14, tri#1, i_red=0")
    for G in graphs.triangulations(14, minimum_degree=5):
        D = dual_of(G)
        D.is_planar(set_embedding=True)
        chosen = None
        for face in D.faces():
            if len(face) != 5:
                continue
            res = apply_reduction(D, face, 0, 9999)
            if res is None:
                continue
            H_1, named_1 = res['H'], res['named']
            edges_1, colorings_1 = proper_3_edge_colorings(H_1)
            cand = [c for c in colorings_1
                    if matches_chord_apex_kempe(edges_1, c, named_1)]
            if cand:
                chosen = (face, res, cand)
                break
        if chosen is None:
            continue
        face, res, cand = chosen
        H_1, named_1 = res['H'], res['named']
        edges_1, _ = proper_3_edge_colorings(H_1)
        if not cand:
            continue
        phi_1 = cand[0]
        phi_1_dict = {frozenset((e[0], e[1])): c for e, c in zip(edges_1, phi_1)}
        H_final, phi_final_dict, all_named = run_algorithm_to_completion(
            H_1, phi_1_dict, named_1)
        edges_f, colorings_f = proper_3_edge_colorings(H_final)
        print(f"  H_t*: |V|={H_final.order()}, |E|={H_final.size()}, "
              f"{len(colorings_f)} proper 3-edge-colorings, "
              f"{len(all_named)} named-edge steps.")

        # Kempe equivalence classes via union-find
        col_idx = {c: i for i, c in enumerate(colorings_f)}
        parent = list(range(len(colorings_f)))

        def find(x):
            while parent[x] != x:
                parent[x] = parent[parent[x]]
                x = parent[x]
            return x

        def union(x, y):
            rx, ry = find(x), find(y)
            if rx != ry:
                parent[rx] = ry

        for c_idx, col in enumerate(colorings_f):
            for pair in [(0, 1), (0, 2), (1, 2)]:
                visited = set()
                for start in range(len(edges_f)):
                    if col[start] not in pair or start in visited:
                        continue
                    cyc = kempe_cycle(edges_f, col, start, pair)
                    visited |= cyc
                    a, b = pair
                    new_col = list(col)
                    for k in cyc:
                        if new_col[k] == a:
                            new_col[k] = b
                        elif new_col[k] == b:
                            new_col[k] = a
                    new_col = tuple(new_col)
                    if new_col != col and new_col in col_idx:
                        union(c_idx, col_idx[new_col])

        roots = {}
        for i in range(len(colorings_f)):
            r = find(i)
            roots.setdefault(r, []).append(i)

        # phi_t* identification
        phi_final = tuple(phi_final_dict[frozenset((e[0], e[1]))]
                          for e in edges_f)
        phi_idx = colorings_f.index(phi_final)
        phi_root = find(phi_idx)

        print()
        print(f"  Found {len(roots)} Kempe equivalence classes:")
        for r, members in sorted(roots.items(), key=lambda kv: -len(kv[1])):
            ad = [m for m in members
                  if is_all_distinct(edges_f, colorings_f[m], all_named)]
            label = " (contains phi_t*)" if r == phi_root else ""
            print(f"    class size {len(members)}: "
                  f"{len(ad)} all-distinct colorings{label}")
        return


if __name__ == '__main__':
    main()