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

"""Investigate the conjecture: H_t* has the same number of Kempe equivalence
classes as H_1.

For each min-deg-5 triangulation G (up to some n) and each reduction index
i_red in {0,...,4}:
  - Build H_1 from the first pentagonal face of G' = dual(G).
  - Count Kempe classes of H_1.
  - Run Algorithm 3.1 to completion with Sage's edge_coloring as phi_1.
  - Count Kempe classes of H_t*.
  - Print a row.

Run with: sage experiments/check_kempe_class_invariance.py
"""
from sage.all import Graph
from sage.graphs.graph_generators import graphs
from sage.graphs.graph_coloring import edge_coloring
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 num_kempe_classes(G):
    """Count the Kempe equivalence classes of proper 3-edge-colorings of G."""
    edges, colorings = proper_3_edge_colorings(G)
    if not colorings:
        return None, 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_coloring_dict, named_step1,
                                vn_start=10000):
    H = H.copy()
    coloring = dict(phi_1_coloring_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, coloring


def main():
    print(f"{'n':>3} {'tri#':>5} {'i_red':>5} | "
          f"{'|V(H_1)|':>9} {'|E(H_1)|':>9} {'#col(H_1)':>10} {'#K(H_1)':>8} | "
          f"{'|V(H_t*)|':>10} {'|E(H_t*)|':>10} {'#col(H_t*)':>11} {'#K(H_t*)':>10}")
    print("-" * 120)
    for n in [12, 14]:
        try:
            triangulations = list(graphs.triangulations(n, minimum_degree=5))
        except Exception as ex:
            print(f"  cannot enumerate n={n}: {ex}")
            continue
        for tri_idx, G in enumerate(triangulations, start=1):
            G_copy = G.copy()
            G_copy.is_planar(set_embedding=True)
            D = dual_of(G_copy)
            D.is_planar(set_embedding=True)
            # iterate over all pentagonal faces (or just the first)
            faces = [f for f in D.faces() if len(f) == 5]
            if not faces:
                continue
            face = faces[0]
            for i_red in range(5):
                res = apply_reduction(D, face, i_red, 9999)
                if res is None:
                    continue
                H_1 = res['H']
                # count Kempe classes of H_1
                n_col_h1, n_kc_h1 = num_kempe_classes(H_1)
                # run algorithm to completion with Sage's edge_coloring as phi_1
                classes = edge_coloring(H_1, value_only=False)
                if len(classes) > 3:
                    continue
                phi_1_dict = {}
                for k, edge_list in enumerate(classes):
                    for u, v in edge_list:
                        phi_1_dict[frozenset([u, v])] = k
                H_final, _ = run_algorithm_to_completion(H_1, phi_1_dict,
                                                        res['named'])
                n_col_hf, n_kc_hf = num_kempe_classes(H_final)
                same = "same" if n_kc_h1 == n_kc_hf else "DIFFERENT"
                print(f"{n:>3} {tri_idx:>5} {i_red:>5} | "
                      f"{H_1.order():>9} {H_1.size():>9} {n_col_h1:>10} {n_kc_h1:>8} | "
                      f"{H_final.order():>10} {H_final.size():>10} "
                      f"{n_col_hf:>11} {n_kc_hf:>10}  {same}")
                sys.stdout.flush()


if __name__ == '__main__':
    main()