coloring_nested_tire_graphs: cut-and-depth-label procedure with Holton-McKay #0 example
Adds a new note describing a cut-and-depth-label procedure for
the dual G' of a maximal planar G:
1. Find a 6-edge cut C in G'.
2. Remove cut edges → G'_0, G'_1.
3. In each G'_i:
a. V_i = degree-2 vertices (vertices incident to exactly 1
cut edge, hence degree 3-1=2 in induced subgraph).
b. For each v ∈ V_i, add a pendant edge to a new vertex.
Label pendants depth 0.
c. BFS-propagate: edges adjacent to a depth-d edge get
depth d+1, until all edges are labelled.
Worked example on Holton-McKay graph #0 (38-vertex non-Hamiltonian
cubic plane graph, dual of a 21-vertex triangulation):
- 128 distinct 6-edge cuts found by greedy search.
- Best matching cut: |S| = 10, cut = 6 edges with 12 distinct
endpoints (6 per side).
- G'_0: 10 + 6 = 16 vertices, max depth 2.
- G'_1: 28 + 6 = 34 vertices, max depth 7.
The procedure mirrors the 4CT cut-and-reglue reducibility scheme:
each G'_i has pendants restoring cubicity at the boundary; the
depth labels organize G'_i into concentric layers by distance to
the cut. This is the dual analogue of plane depth from a level
cycle (cf. the level-cycle generalization discussion).
Files:
experiments/cut_depth_label.py
notes/cut_depth_label.tex (3 pages)
notes/fig_cut_depth_label.png
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
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"""Cut-and-depth-label procedure on a Holton-McKay graph.
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Procedure:
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1. Take G' (we use a Holton-McKay non-Hamiltonian 38-vertex cubic
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plane graph, viewed as the dual of a 21-vertex triangulation G).
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2. Find 6 cut edges of G' whose removal disconnects it into two
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non-trivial cubic-minus-boundary pieces G'_0 and G'_1.
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3. In each G'_i:
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a. V_i = vertices of degree 2 (the 6 endpoints in G'_i of cut
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edges).
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b. For each v ∈ V_i, add a new pendant edge to a new vertex.
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Label these pendant edges depth 0.
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c. BFS-propagate: for d = 0, 1, ..., label every edge sharing
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a vertex with a depth-d edge (and not yet labelled) as
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depth d + 1.
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d. Stop when every edge has a depth label.
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4. Render G'_0 and G'_1 with edges coloured by depth.
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"""
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import os
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import sys
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from collections import deque, defaultdict
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import math
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import matplotlib.pyplot as plt
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import matplotlib.colors as mcolors
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import numpy as np
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from sage.all import Graph
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HM_FILE = ('/Users/didericis/Code/math-research/papers/'
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'even_level_graph_generators/experiments/nonham38m4.pc')
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def parse_planar_code(path):
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with open(path, 'rb') as f:
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data = f.read()
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header = b'>>planar_code<<'
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assert data.startswith(header), data[:20]
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pos = len(header)
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sage_graphs = []
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while pos < len(data):
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n = data[pos]; pos += 1
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edges = set()
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for v in range(n):
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while True:
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w = data[pos]; pos += 1
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if w == 0:
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break
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edges.add(frozenset((v, w - 1)))
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G = Graph([tuple(e) for e in edges], multiedges=False, loops=False)
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G.is_planar(set_embedding=True)
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sage_graphs.append(G)
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return sage_graphs
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def find_six_edge_cut(G, prefer_balanced=True, prefer_matching=True):
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"""Find a 6-edge cut: a set of 6 edges whose removal yields two
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non-trivial components.
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For a cubic graph: edges between S and complement
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= 3|S| - 2 * e(S). We want this to equal 6.
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Search strategy: BFS-grow a vertex set S from each possible start,
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track the boundary edge count. Collect all sizes that yield a
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6-edge cut. Return the best one according to:
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prefer_matching: cut is a "matching cut" (each boundary vertex
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has exactly one cut edge).
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prefer_balanced: |S| closest to n/2.
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"""
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n = G.order()
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candidates = []
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for start in G.vertices():
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S = {start}
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boundary = 3
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# BFS-grow; at each step pick the frontier vertex that adds the
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# most internal edges (reduces boundary).
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prev_size = -1
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while True:
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if boundary == 6 and 2 <= len(S) <= n - 2:
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cut = [(u, v) for u in S
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for v in G.neighbors(u) if v not in S]
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if len(cut) == 6:
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candidates.append((frozenset(S), cut))
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if len(S) == prev_size:
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break
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prev_size = len(S)
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# frontier
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frontier = []
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for w in G.vertices():
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if w in S: continue
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if any(nb in S for nb in G.neighbors(w)):
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edges_into_S = sum(1 for nb in G.neighbors(w) if nb in S)
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frontier.append((edges_into_S, w))
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if not frontier:
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break
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frontier.sort(reverse=True)
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_, w = frontier[0]
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edges_into_S = sum(1 for nb in G.neighbors(w) if nb in S)
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S.add(w)
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boundary = boundary + 3 - 2 * edges_into_S
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if len(S) >= n - 1:
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break
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if not candidates:
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return None, None
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# Pick best: prefer matching cuts (both sides have 6 distinct
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# boundary vertices = 6 cut edges form a matching).
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def score(item):
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S, cut = item
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endpoints_in_S = [u for (u, v) in cut if u in S] + \
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[v for (u, v) in cut if v in S]
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endpoints_in_Sc = [u for (u, v) in cut if u not in S] + \
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[v for (u, v) in cut if v not in S]
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is_matching = (len(set(endpoints_in_S)) == 6 and
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len(set(endpoints_in_Sc)) == 6)
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# Distinctness within each side
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deg2_S = len(set(endpoints_in_S))
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deg2_Sc = len(set(endpoints_in_Sc))
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balance = abs(len(S) - n / 2)
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# Prefer maximum total V coverage, then matching, then balanced
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return (-(deg2_S + deg2_Sc), -int(is_matching), balance)
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candidates.sort(key=score)
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# Dedupe identical S sets
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seen = set()
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unique = []
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for S, cut in candidates:
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if S in seen: continue
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seen.add(S)
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unique.append((S, cut))
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print(f' Found {len(unique)} distinct 6-edge cuts')
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return unique[0]
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def apply_procedure(G, S, cut, side_label='0'):
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"""Build G'_i from G[S] plus pendant edges at degree-2 vertices.
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Per the user's procedure:
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a. V = vertices of degree 2 in the induced subgraph (i.e.,
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original cubic vertices that have exactly 1 cut edge,
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hence degree 3 - 1 = 2 in the induced subgraph).
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b. Add 1 pendant edge per v in V.
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c. Label pendants depth 0, BFS-propagate.
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Returns (graph, pos, edge_depths, deg2_vertices, pendant_map,
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high_deg_loss). high_deg_loss tracks vertices with >= 2 cut
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edges that don't receive pendants under the strict procedure.
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"""
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induced_edges = [(u, v) for (u, v) in G.edges(labels=False)
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if u in S and v in S]
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H = Graph(induced_edges, multiedges=False, loops=False)
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# Ensure all vertices of S are present (some may be isolated)
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for v in S:
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H.add_vertex(v)
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# Compute degree of each vertex in induced subgraph
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induced_deg = {v: H.degree(v) for v in S}
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# V = degree-2 vertices
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V_deg2 = sorted([v for v in S if induced_deg[v] == 2])
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high_loss = sorted([v for v in S if induced_deg[v] < 2])
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pendant_edges = []
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next_pendant_id = (max(G.vertices()) + 1)
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pendant_to_boundary = {}
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for v in V_deg2:
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H.add_edge(v, next_pendant_id)
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pendant_edges.append((min(v, next_pendant_id),
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max(v, next_pendant_id)))
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pendant_to_boundary[next_pendant_id] = v
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next_pendant_id += 1
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# BFS-label edges by depth
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edge_depth = {}
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queue = deque()
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for e in pendant_edges:
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edge_depth[e] = 0
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queue.append((e, 0))
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while queue:
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(u, v), d = queue.popleft()
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# adjacent edges share vertex u or v
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for endpoint in (u, v):
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for nb in H.neighbors(endpoint):
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e_nb = (min(endpoint, nb), max(endpoint, nb))
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if e_nb == (u, v): continue
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if e_nb not in edge_depth:
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edge_depth[e_nb] = d + 1
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queue.append((e_nb, d + 1))
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# Layout: use Sage's planar embedding
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pos = H.layout(layout='planar') if H.is_planar() else H.layout()
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return H, pos, edge_depth, V_deg2, pendant_to_boundary, high_loss
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def draw_labeled_graph(ax, H, pos, edge_depth, boundary_vertices,
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pendant_to_boundary, title):
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max_depth = max(edge_depth.values()) if edge_depth else 0
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cmap = plt.get_cmap('viridis', max_depth + 1)
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# Draw edges by depth
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legend_handles = []
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for d in range(max_depth + 1):
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color = cmap(d / max(max_depth, 1))
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for e, ed in edge_depth.items():
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if ed != d: continue
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u, v = e
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(x1, y1) = pos[u]
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(x2, y2) = pos[v]
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ax.plot([x1, x2], [y1, y2], color=color,
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linewidth=2.0 + (1.5 if d == 0 else 0),
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linestyle=('--' if d == 0 else '-'),
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zorder=1)
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legend_handles.append(
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plt.Line2D([], [], color=color, linewidth=2.5,
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linestyle='--' if d == 0 else '-',
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label=f'depth {d}')
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)
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# Draw vertices
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for v, (x, y) in pos.items():
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if v in pendant_to_boundary:
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ax.plot(x, y, 's', color='#ffaa66', markersize=8, zorder=2,
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markeredgecolor='#aa5500')
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elif v in boundary_vertices:
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ax.plot(x, y, 'o', color='#ff5555', markersize=10, zorder=2,
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markeredgecolor='#aa0000')
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else:
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ax.plot(x, y, 'o', color='#888', markersize=6, zorder=2)
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ax.set_aspect('equal')
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ax.axis('off')
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ax.set_title(title, fontsize=11)
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ax.legend(handles=legend_handles, loc='upper left',
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bbox_to_anchor=(1.02, 1.0), fontsize=8, frameon=False)
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def main():
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here = os.path.dirname(os.path.abspath(__file__))
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out = os.path.normpath(os.path.join(here, '..', 'notes',
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'fig_cut_depth_label.png'))
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gs = parse_planar_code(HM_FILE)
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print(f'Loaded {len(gs)} Holton-McKay graphs')
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# Pick the first one
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G = gs[0]
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print(f'Graph 0: {G.order()} vertices, {G.size()} edges, '
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f'cubic={all(d == 3 for d in G.degree())}, planar={G.is_planar()}')
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# Find a 6-edge cut
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S, cut = find_six_edge_cut(G)
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if S is None:
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print('No 6-edge cut found by greedy search.')
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return
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print(f'Found 6-edge cut: |S| = {len(S)}, |cut| = {len(cut)}')
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print(f'Cut edges: {cut}')
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S0 = frozenset(S)
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S1 = frozenset(G.vertices()) - S0
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H0, pos0, ed0, bv0, pmap0, hl0 = apply_procedure(G, S0, cut, '0')
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H1, pos1, ed1, bv1, pmap1, hl1 = apply_procedure(G, S1, cut, '1')
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print(f"G'_0: {H0.order()} vertices ({len(S0)} original + "
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f'{len(pmap0)} pendant), {H0.size()} edges')
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print(f" V (degree-2 vertices receiving pendants): {len(bv0)} vertices")
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print(f" Multi-cut vertices not in V: {hl0}")
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print(f' Max depth: {max(ed0.values())}')
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print(f"G'_1: {H1.order()} vertices ({len(S1)} original + "
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f'{len(pmap1)} pendant), {H1.size()} edges')
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print(f" V (degree-2 vertices receiving pendants): {len(bv1)} vertices")
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print(f" Multi-cut vertices not in V: {hl1}")
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print(f' Max depth: {max(ed1.values())}')
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# Draw
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fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(16, 8.0))
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draw_labeled_graph(ax0, H0, pos0, ed0, bv0, pmap0,
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f"$G'_0$ (|S| = {len(S0)}, |V| = {len(bv0)}, "
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f"max depth = {max(ed0.values())})")
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draw_labeled_graph(ax1, H1, pos1, ed1, bv1, pmap1,
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f"$G'_1$ (|S| = {len(S1)}, |V| = {len(bv1)}, "
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f"max depth = {max(ed1.values())})")
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fig.suptitle("Cut-and-depth-label procedure on Holton-McKay graph #0 "
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"(38 vertices, cubic, planar, non-Hamiltonian)\n"
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"Dashed orange = pendant edges (depth 0); colored by "
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"BFS depth from pendants in line-graph sense",
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fontsize=11, y=1.00)
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plt.tight_layout()
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plt.savefig(out, dpi=160, bbox_inches='tight')
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plt.close()
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print(f'Wrote {out}')
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if __name__ == '__main__':
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main()
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@@ -0,0 +1,8 @@
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\relax
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\@writefile{toc}{\contentsline {paragraph}{The chosen 6-edge cut.}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{Resulting half-graphs.}{2}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{2}{}\protected@file@percent }
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\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
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\newlabel{fig:cut-depth-label}{{1}{2}}
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\@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent }
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\gdef \@abspage@last{3}
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entering extended mode
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restricted \write18 enabled.
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**cut_depth_label.tex
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n[]level[]graph[]generators/experiments/nonham38m4.pc\OT1/cmr/m/n/10.95 ).
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\OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O
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T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo
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\documentclass[11pt]{article}
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\usepackage{amsmath,amssymb,amsthm}
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\usepackage{graphicx}
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\usepackage{geometry}
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\usepackage{booktabs}
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\usepackage{caption}
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\geometry{margin=1in}
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\title{Cut-and-depth-label: a procedure for labelling half-graphs of\\
|
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a $6$-edge cut by ``distance to the cut''}
|
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\author{}
|
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\date{}
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\newtheorem*{obs}{Observation}
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\newtheorem*{prop}{Proposition}
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\begin{document}
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\maketitle
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\section*{Procedure}
|
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Given a maximal planar graph $G$ and its dual $G'$:
|
||||
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\begin{enumerate}
|
||||
\item Find a $6$-edge cut $C \subseteq E(G')$ partitioning
|
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$V(G')$ into $S$ and $V \setminus S$ with both sides
|
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non-empty.
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\item Remove the cut edges to obtain two graphs
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$G'_0 := G'[S] \cup \emptyset_C$ and
|
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$G'_1 := G'[V \setminus S] \cup \emptyset_C$ (each a cubic
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graph minus boundary edges).
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\item In each $G'_i$:
|
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\begin{enumerate}
|
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\item Let $V_i$ be the set of vertices of degree $2$ in
|
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$G'_i$ (i.e.\ original cubic vertices incident to
|
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exactly one cut edge; the cut edge accounts for the
|
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missing edge, so the vertex sits at degree $3 - 1 =
|
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2$).
|
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\item For each $v \in V_i$, attach a new pendant edge to
|
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a fresh vertex, and \textbf{label these pendant
|
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edges with depth $0$}.
|
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\item For $d = 0, 1, 2, \ldots$: label every unlabelled
|
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edge that shares a vertex with a depth-$d$ edge with
|
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depth $d + 1$.
|
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\item Stop when every edge has a depth label.
|
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\end{enumerate}
|
||||
\end{enumerate}
|
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\noindent\textbf{Interpretation.} The depth labels give a BFS
|
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distance in the line graph of $G'_i$ starting from the pendants
|
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added at the cut. Equivalently, the depth of an edge $e$ in $G'_i$
|
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is the minimum number of edges traversed (via shared-vertex
|
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adjacency) to reach a pendant.
|
||||
|
||||
\noindent\textbf{Caveat.} If the cut $C$ is not a \emph{matching}
|
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cut (i.e., the $6$ cut edges share vertices), then some boundary
|
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vertices have degree $< 2$ in $G'_i$ and do not receive a pendant
|
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under the strict reading of step~3(a). When the cut is a
|
||||
matching, each of the $12$ boundary vertices ($6$ per side) has
|
||||
degree exactly $2$ and receives a pendant; the construction is
|
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symmetric.
|
||||
|
||||
\section*{Example: Holton-McKay graph \#0}
|
||||
|
||||
We apply the procedure to the first of the six non-Hamiltonian
|
||||
$38$-vertex cubic plane graphs found by Holton and McKay (loaded
|
||||
from \texttt{papers/even\_level\_graph\_generators/experiments/nonham38m4.pc}).
|
||||
This $G'$ is itself a cubic plane graph; its dual $G$ is a
|
||||
$21$-vertex triangulation.
|
||||
|
||||
\paragraph{The chosen 6-edge cut.}
|
||||
Greedy search over $128$ distinct $6$-edge cuts in $G'$, preferring
|
||||
\emph{matching cuts} (both sides have $6$ distinct boundary
|
||||
vertices) and then balanced $|S|$, returns
|
||||
\[
|
||||
|S| = 10, \qquad
|
||||
C = \{(34, 29),\, (35, 30),\, (26, 22),\, (27, 23),\,
|
||||
(28, 24),\, (31, 25)\}.
|
||||
\]
|
||||
This is a matching cut: the $6$ edges have $12$ distinct endpoints,
|
||||
$6$ on each side.
|
||||
|
||||
\paragraph{Resulting half-graphs.}
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{lcc}
|
||||
\toprule
|
||||
& $G'_0$ & $G'_1$ \\
|
||||
\midrule
|
||||
$|S|$ & $10$ & $28$ \\
|
||||
Original vertices in $G'_i$ & $10$ & $28$ \\
|
||||
$|V_i|$ (pendants added) & $6$ & $6$ \\
|
||||
Total vertices in $G'_i$ & $16$ & $34$ \\
|
||||
Total edges & $18$ & $45$ \\
|
||||
Max depth assigned & $2$ & $7$ \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
Multi-cut vertices (degree $< 2$ in induced subgraph): \emph{none}
|
||||
on either side, since the cut is a matching.
|
||||
|
||||
\paragraph{Visualization.}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{fig_cut_depth_label.png}
|
||||
\caption{Cut-and-depth-label procedure on Holton-McKay graph \#0
|
||||
($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The
|
||||
$6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$
|
||||
(left) and a $28$-vertex piece $G'_1$ (right). Pendant edges
|
||||
(dashed, dark purple) are at depth $0$; the colour gradient
|
||||
encodes depth $0$ through max depth via BFS in the line-graph
|
||||
sense. Orange squares are the new pendant vertices; red circles
|
||||
are the boundary vertices in $V_i$; gray circles are interior
|
||||
vertices.}
|
||||
\label{fig:cut-depth-label}
|
||||
\end{figure}
|
||||
|
||||
\section*{Connection to chain pigeonhole / 4CT reducibility}
|
||||
|
||||
The procedure mirrors the $4$CT cut-and-reglue scheme
|
||||
(\texttt{rainbow\_proof.tex}, \texttt{worst\_case\_proof\_sketch.tex},
|
||||
\texttt{two\_approaches\_comparison.tex}) at the structural level.
|
||||
After cutting, each $G'_i$ is a cubic-minus-boundary graph; the
|
||||
pendant additions formally restore cubicity at the $6$ degree-$2$
|
||||
boundary vertices. The depth label on each edge measures its
|
||||
``distance to the cut.''
|
||||
|
||||
For a minimum counterexample to the 4CT (i.e., a cubic plane $G'$
|
||||
with no proper $3$-edge-colouring), the depth labels organise each
|
||||
$G'_i$ into concentric layers indexed by distance to the cut. The
|
||||
$3$-edge-colourings of $G'_i$ must extend a colouring at the
|
||||
depth-$0$ pendants (= a ring colouring at the cut); the BFS
|
||||
ordering by depth is the natural induction order for propagating
|
||||
the colouring inward.
|
||||
|
||||
In the tire framework, the cut cycle $\gamma$ in the primal $G$
|
||||
(corresponding to the $6$-edge cut in $G'$ via planar duality) plays
|
||||
the role of the tire's inner boundary on one side and outer boundary
|
||||
on the other. The depth label on $G'_i$-edges is exactly the dual
|
||||
analogue of plane depth from $\gamma$ (cf.\ the level-cycle
|
||||
generalization discussion in the recent conversation).
|
||||
|
||||
\section*{Limitations of this example}
|
||||
|
||||
\begin{itemize}
|
||||
\item Holton-McKay graphs are cyclically $5$-edge-connected (not
|
||||
$6$-edge-connected), so $6$-edge cuts are not the minimum
|
||||
cyclic cut. The smallest cyclic edge cuts in this graph are
|
||||
size $5$.
|
||||
\item The matching $6$-cut found is highly imbalanced ($|S|=10$
|
||||
vs.\ $|S^c|=28$). Searching among the $128$ distinct
|
||||
$6$-edge cuts for a balanced matching cut may give better
|
||||
examples.
|
||||
\item Depth labels propagate via the line graph, not via vertex
|
||||
BFS. An alternative procedure would label \emph{vertices}
|
||||
by BFS distance from the boundary; both yield similar
|
||||
layered structures but with slightly different counts.
|
||||
\end{itemize}
|
||||
|
||||
\end{document}
|
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