"""Investigate the conjecture:
   #(Kempe classes of H_t*) <= #(Kempe classes of H_1).

For each min-deg-5 triangulation G and each (face, i_red) choice giving a
valid H_1, vary phi_1 across many proper 3-edge-colorings of H_1 -- so we
exercise multiple algorithm executions, each potentially producing a
different H_t*. For each execution, count Kempe classes of H_t* and compare
to H_1.

Reports any case where #classes(H_t*) > #classes(H_1) (counterexample to
the conjecture).

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


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 num_kempe_classes(G):
    edges, colorings = proper_3_edge_colorings(G)
    if not colorings:
        return 0, 0
    n = len(colorings)
    parent = list(range(n))

    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

    col_idx = {c: i for i, c in enumerate(colorings)}
    for c_idx, col in enumerate(colorings):
        for pair in [(0, 1), (0, 2), (1, 2)]:
            visited = set()
            for start in range(len(edges)):
                if col[start] not in pair or start in visited:
                    continue
                cyc = kempe_cycle(edges, 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])
    classes = len({find(i) for i in range(n)})
    return n, classes


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)
    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]
        E |= {frozenset(spike), frozenset(side_0), frozenset(side_1),
              frozenset(merged)}
    return H


def main(max_n=15, time_budget_sec=600):
    start = time.time()
    violations = []
    for n in range(12, max_n + 1):
        if time.time() - start > time_budget_sec:
            print(f"Time budget exhausted at n={n}.")
            break
        print(f"\n=== 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):
            if time.time() - start > time_budget_sec:
                break
            G_copy = G.copy()
            G_copy.is_planar(set_embedding=True)
            D = dual_of(G_copy)
            D.is_planar(set_embedding=True)
            for face in D.faces():
                if len(face) != 5:
                    continue
                for i_red in range(5):
                    if time.time() - start > time_budget_sec:
                        break
                    res = apply_reduction(D, face, i_red, 9999)
                    if res is None:
                        continue
                    H_1 = res['H']
                    n_col_h1, kc_h1 = num_kempe_classes(H_1)
                    if kc_h1 == 0:
                        continue
                    # enumerate all proper colorings of H_1 to vary phi_1
                    edges_1, colorings_1 = proper_3_edge_colorings(H_1)
                    # try EVERY phi_1 (might be slow for large H_1)
                    max_kc_hf = 0
                    h_final_seen = {}  # signature -> classes
                    for phi_1 in colorings_1:
                        phi_1_dict = {frozenset((e[0], e[1])): c
                                      for e, c in zip(edges_1, phi_1)}
                        H_final = run_algorithm_to_completion(
                            H_1, phi_1_dict, res['named'])
                        # signature = (|V|, |E|, sorted edge tuples)
                        sig = (H_final.order(), H_final.size())
                        if sig in h_final_seen:
                            continue
                        _, kc_hf = num_kempe_classes(H_final)
                        h_final_seen[sig] = kc_hf
                        if kc_hf > max_kc_hf:
                            max_kc_hf = kc_hf
                    over = max_kc_hf > kc_h1
                    flag = "*** OVER ***" if over else "ok"
                    print(f"  n={n} tri#{tri_idx} face|{len(face)} i_red={i_red}: "
                          f"H_1 kc={kc_h1}, max H_t* kc over {len(h_final_seen)} "
                          f"distinct outputs = {max_kc_hf} {flag}")
                    if over:
                        violations.append((n, tri_idx, i_red, kc_h1, max_kc_hf))
                    sys.stdout.flush()
                break  # first face only

    print()
    if violations:
        print(f"Counterexamples (#K(H_t*) > #K(H_1)): {len(violations)}")
        for v in violations:
            print(f"  {v}")
    else:
        print("No counterexamples found within tested range.")


if __name__ == '__main__':
    main()