Record level-cycle coloring conjectures
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"""Empirical test for the level-cycle three-colour conjecture.
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The weakened conjecture says: for every maximal planar graph G, there
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is some level source S and some proper 4-vertex-colouring c such that
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every simple cycle contained in a single level G[L_d] uses at most
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three colours.
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Important: this checks the cycle-by-cycle version. Two cycles in the
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same level or the same inner outerplanar component may omit different
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colours.
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Run examples:
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sage -python experiments/check_level_cycle_three_color.py 4 9
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sage -python experiments/check_level_cycle_three_color.py 4 8 --quantifier all-sources
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sage -python experiments/check_level_cycle_three_color.py 4 8 --sources all
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"""
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from __future__ import annotations
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import argparse
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from collections import deque
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from itertools import combinations
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from typing import Any, Iterable, Iterator, Sequence, cast
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from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module
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from sage.graphs.graph_coloring import all_graph_colorings # type: ignore[attr-defined] # pylint: disable=no-name-in-module
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Coloring = dict[Any, int]
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Source = tuple[Any, ...]
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def vertex_key(v: Any) -> str:
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"""Stable ordering key for Sage vertex labels."""
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return repr(v)
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def is_induced_cycle(g: Graph, vertices: Sequence[Any]) -> bool:
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"""Return True iff G[vertices] is a simple cycle."""
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if len(vertices) < 3:
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return False
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h = cast(Graph, g.subgraph(list(vertices)))
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return h.is_connected() and h.num_edges() == len(vertices) and all(
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h.degree(v) == 2 for v in h.vertices()
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)
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def induced_cycle_sources(g: Graph, max_size: int | None = None) -> Iterator[Source]:
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"""Yield every vertex set inducing a simple cycle in G."""
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vertices = sorted(g.vertices(), key=vertex_key)
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upper = len(vertices) if max_size is None else min(max_size, len(vertices))
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for k in range(3, upper + 1):
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for subset in combinations(vertices, k):
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if is_induced_cycle(g, subset):
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yield tuple(subset)
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def level_sources(g: Graph, mode: str, max_cycle_source_size: int | None) -> Iterator[Source]:
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"""Yield level sources according to the requested mode."""
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if mode in ("vertex", "all"):
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for v in sorted(g.vertices(), key=vertex_key):
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yield (v,)
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if mode in ("cycle", "all"):
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yield from induced_cycle_sources(g, max_cycle_source_size)
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def distances_from_source(g: Graph, source: Source) -> dict[Any, int]:
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"""Shortest-path distance from a vertex or cycle source."""
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if len(source) == 1:
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return dict(g.shortest_path_lengths(source[0]))
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distances = {v: 0 for v in source}
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queue: deque[Any] = deque(source)
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while queue:
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v = queue.popleft()
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for w in g.neighbor_iterator(v):
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if w in distances:
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continue
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distances[w] = distances[v] + 1
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queue.append(w)
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return distances
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def simple_cycle_vertex_sets(g: Graph) -> set[frozenset[Any]]:
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"""Enumerate vertex sets of simple cycles in an undirected graph.
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The DFS only starts a cycle at its least vertex under vertex_key, which
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avoids most duplicates; the final frozenset de-duplicates cycles with the
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same vertex set.
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"""
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vertices = sorted(g.vertices(), key=vertex_key)
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index = {v: i for i, v in enumerate(vertices)}
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cycles: set[frozenset[Any]] = set()
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def dfs(start: Any, current: Any, path: list[Any], seen: set[Any]) -> None:
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for nxt in g.neighbor_iterator(current):
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if nxt == start:
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if len(path) >= 3:
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cycles.add(frozenset(path))
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continue
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if nxt in seen:
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continue
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if index[nxt] <= index[start]:
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continue
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seen.add(nxt)
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path.append(nxt)
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dfs(start, nxt, path, seen)
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path.pop()
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seen.remove(nxt)
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for start in vertices:
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dfs(start, start, [start], {start})
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return cycles
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def level_cycle_violation(
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g: Graph, distances: dict[Any, int], coloring: Coloring
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) -> tuple[int, frozenset[Any], set[int]] | None:
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"""Return the first level cycle using all four colours, if any."""
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by_level: dict[int, list[Any]] = {}
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for v, d in distances.items():
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by_level.setdefault(d, []).append(v)
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for d in sorted(by_level):
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if len(by_level[d]) < 3:
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continue
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h = cast(Graph, g.subgraph(by_level[d]))
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for cycle in simple_cycle_vertex_sets(h):
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used = {coloring[v] for v in cycle}
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if len(used) > 3:
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return d, cycle, used
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return None
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def coloring_witness(
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g: Graph,
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source: Source,
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max_colorings: int | None,
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) -> tuple[Coloring | None, int, bool]:
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"""Find a proper 4-colouring satisfying the conjectured restriction.
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Returns (witness, checked_count, exhausted). If witness is None and
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exhausted is False, max_colorings was reached before a decision.
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"""
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distances = distances_from_source(g, source)
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checked = 0
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for raw in all_graph_colorings(g, 4, vertex_color_dict=True):
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coloring = cast(Coloring, raw)
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checked += 1
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if level_cycle_violation(g, distances, coloring) is None:
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return coloring, checked, True
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if max_colorings is not None and checked >= max_colorings:
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return None, checked, False
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return None, checked, True
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def source_label(source: Source) -> str:
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if len(source) == 1:
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return f"vertex:{source[0]}"
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return "cycle:{" + ",".join(str(v) for v in source) + "}"
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def test_graph(
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g: Graph,
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sources: Iterable[Source],
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max_colorings: int | None,
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stop_first: bool,
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quantifier: str,
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) -> tuple[bool, bool, int]:
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"""Test a graph over selected sources.
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Returns (passed, complete, sources_checked).
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"""
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checked_sources = 0
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complete = True
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found_any_source = False
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for source in sources:
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checked_sources += 1
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witness, n_checked, exhausted = coloring_witness(g, source, max_colorings)
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if witness is None:
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if not exhausted:
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complete = False
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status = "FAIL" if exhausted else "UNKNOWN"
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print(
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f" {status}: source={source_label(source)}, "
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f"colorings_checked={n_checked}"
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)
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if exhausted and quantifier == "all-sources":
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distances = distances_from_source(g, source)
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first = next(all_graph_colorings(g, 4, vertex_color_dict=True), None)
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if first is not None:
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violation = level_cycle_violation(g, distances, cast(Coloring, first))
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print(f" first_coloring_violation={violation}")
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return False, complete, checked_sources
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if stop_first:
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if quantifier == "all-sources":
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return True, False, checked_sources
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if not exhausted:
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return True, False, checked_sources
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else:
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found_any_source = True
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print(
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f" pass: source={source_label(source)}, "
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f"colorings_checked={n_checked}"
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)
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if quantifier == "exists-source":
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return True, complete, checked_sources
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if quantifier == "exists-source" and not found_any_source:
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return False, complete, checked_sources
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return True, complete, checked_sources
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def parse_args() -> argparse.Namespace:
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parser = argparse.ArgumentParser(description=__doc__)
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parser.add_argument("n_min", type=int, nargs="?", default=4)
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parser.add_argument("n_max", type=int, nargs="?", default=9)
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parser.add_argument(
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"--sources",
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choices=("vertex", "cycle", "all"),
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default="vertex",
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help=(
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"which candidate level sources to search; cycle means induced "
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"cycle sources"
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),
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)
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parser.add_argument(
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"--quantifier",
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choices=("exists-source", "all-sources"),
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default="exists-source",
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help=(
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"exists-source tests the weakened conjecture; all-sources tests "
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"the stronger earlier version"
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),
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)
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parser.add_argument(
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"--max-cycle-source-size",
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type=int,
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default=None,
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help="optional cap on induced cycle source size",
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)
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parser.add_argument(
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"--max-colorings",
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type=int,
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default=None,
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help="optional cap per graph/source; capped searches report UNKNOWN",
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)
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parser.add_argument(
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"--full",
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action="store_true",
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help="continue after failures/unknowns instead of stopping at first",
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)
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return parser.parse_args()
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def main() -> int:
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args = parse_args()
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stop_first = not args.full
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total_graphs = 0
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total_sources = 0
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unknown = 0
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for n in range(args.n_min, args.n_max + 1):
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print(f"=== n={n} ===")
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for idx, g in enumerate(graphs.triangulations(n), start=1):
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total_graphs += 1
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source_list = list(
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level_sources(g, args.sources, args.max_cycle_source_size)
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)
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print(f" graph #{idx}: sources={len(source_list)}")
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passed, complete, checked_sources = test_graph(
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g, source_list, args.max_colorings, stop_first, args.quantifier
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)
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total_sources += checked_sources
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if not complete:
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unknown += 1
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if not passed:
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print(
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f"COUNTEREXAMPLE candidate: n={n}, graph_index={idx}, "
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f"source_mode={args.sources}, quantifier={args.quantifier}"
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)
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print(f" edges={sorted(tuple(sorted(e)) for e in g.edges(labels=False))}")
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return 1
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if not complete and stop_first:
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print(
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f"UNKNOWN: n={n}, graph_index={idx}, "
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f"source_mode={args.sources}, quantifier={args.quantifier}"
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)
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return 2
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print(
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f"PASS: checked {total_graphs} triangulations and {total_sources} "
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f"sources; unknown_graphs={unknown}"
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)
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return 0 if unknown == 0 else 2
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if __name__ == "__main__":
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raise SystemExit(main())
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@@ -38,12 +38,15 @@
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\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}}
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\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.23\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{16}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.23\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{16}{}\protected@file@percent }
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\newlabel{fig:tire-tree-decomposition}{{5}{16}}
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\newlabel{fig:tire-tree-decomposition}{{5}{16}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.25}{16}}
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\newlabel{rem:level-cycle-motivation}{{1.25}{16}}
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\newlabel{conj:level-cycle-three-colour}{{1.26}{16}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{16}}
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\newlabel{def:seam}{{1.27}{16}}
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\newlabel{conj:false-universal-level-cycle-three-colour}{{1.27}{17}}
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\newlabel{def:partial-tire-tree}{{1.28}{17}}
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\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{17}}
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\newlabel{lem:seam-edge-shared}{{1.29}{17}}
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\newlabel{conj:level-cycle-three-colour}{{1.29}{17}}
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\newlabel{conj:seam-counterexample}{{1.30}{17}}
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\newlabel{def:seam}{{1.30}{17}}
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\newlabel{def:partial-tire-tree}{{1.31}{18}}
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\newlabel{lem:seam-edge-shared}{{1.32}{18}}
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\newlabel{conj:seam-counterexample}{{1.33}{18}}
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\bibcite{tait-original}{1}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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@@ -52,5 +55,5 @@
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{18}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{19}{}\protected@file@percent }
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\gdef \@abspage@last{18}
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\gdef \@abspage@last{19}
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@@ -1,5 +1,6 @@
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# Fdb version 3
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# Fdb version 3
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["pdflatex"] 1780292322 "paper.tex" "paper.pdf" "paper" 1780292323
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[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr
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program for tire treads.
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program for tire treads.
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\end{remark}
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\end{remark}
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\begin{remark}[Motivation for level-cycle restrictions]
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\label{rem:level-cycle-motivation}
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The tire-tree decomposition reduces global colouring questions to local
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cycles. Without further structure, the number of boundary colour states
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in principle carry any proper restriction of a $4$-colouring. The
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\end{remark}
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\begin{definition}[Level-cycle three-colour restriction]
|
\begin{definition}[Level-cycle three-colour restriction]
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\label{def:level-cycle-three-colour-restriction}
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\label{def:level-cycle-three-colour-restriction}
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Let $G$ be a maximal planar graph, let $S \subseteq V(G)$ be a level
|
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@@ -1276,13 +1291,51 @@ tread, or the same inner outerplanar component are not required to
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omit the same colour.
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omit the same colour.
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\end{definition}
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\end{definition}
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\begin{conjecture}[Level-cycle three-colour conjecture]
|
\begin{conjecture}[False universal-source form]
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\label{conj:level-cycle-three-colour}
|
\label{conj:false-universal-level-cycle-three-colour}
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Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any
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Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any
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level source. Then $G$ admits a proper $4$-vertex-colouring with the
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level source. Then $G$ admits a proper $4$-vertex-colouring with the
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level-cycle three-colour restriction with respect to $S$.
|
level-cycle three-colour restriction with respect to $S$.
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\end{conjecture}
|
\end{conjecture}
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\begin{example}[Counterexample to Conjecture~\ref{conj:false-universal-level-cycle-three-colour}]
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\label{ex:universal-level-cycle-counterexample}
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$\{1,2,3,4,5,6,7,8\}$ with edge set
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\[
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\begin{aligned}
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E(G)=\{&
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12,13,14,15,16,17,23,26,27,34,35,36,38,\\
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\]
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\[
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L_0=\{7\},\qquad L_1=\{1,2,6\},\qquad
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\]
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four distinct colours. The edges $34$, $45$, $58$, $38$, and $35$
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force all pairs among $\{3,4,5,8\}$ except possibly $\{4,8\}$ to have
|
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distinct colours. If $4$ and $8$ had the same colour, then vertex $6$,
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which is adjacent to $3$, $5$, and $8$, would have to use the fourth
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colour; but vertex $1$ is adjacent to $3$, $4$, $5$, and $6$, and
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would then be adjacent to all four colours, impossible in a proper
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$4$-colouring. Hence $4$ and $8$ also have distinct colours, so the
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level cycle $(3,4,5,8)$ uses all four colours in every proper
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$4$-colouring of $G$. Therefore no proper $4$-colouring has the
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level-cycle three-colour restriction with respect to $S=\{7\}$.
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|
\end{example}
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|
|
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|
\begin{conjecture}[Level-cycle three-colour conjecture]
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|
\label{conj:level-cycle-three-colour}
|
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|
Let $G$ be a maximal planar graph. Then there exists a level source
|
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$S \subseteq V(G)$ such that $G$ admits a proper $4$-vertex-colouring
|
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|
with the level-cycle three-colour restriction with respect to $S$.
|
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|
\end{conjecture}
|
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|
|
||||||
\begin{definition}[Seam]
|
\begin{definition}[Seam]
|
||||||
\label{def:seam}
|
\label{def:seam}
|
||||||
A \emph{seam} of a maximal planar graph $G$ is a simple cycle
|
A \emph{seam} of a maximal planar graph $G$ is a simple cycle
|
||||||
|
|||||||
Reference in New Issue
Block a user