Redraw n=21 witness figures as crossing-free planar graphs
Replace the radial (crossing-heavy) figure with two crossing-free planar
drawings (networkx planar_layout / Chrobak-Payne):
fig:n21-elgs -- the six witness Even Level Graphs, parity-coloured, with
the bridge-switch-flipped edges dashed red;
fig:n21-duals -- the six resulting duals, with the introduced bridge edges
solid green.
ELG and dual are drawn with independent planar layouts so neither has any
edge crossing (a flip diagonal would otherwise cross other edges when its
quadrilateral is non-convex, which happens for duals 0 and 3). Drop forced
equal aspect so panels fill and labels separate.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -1,12 +1,16 @@
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"""Draw each of the six Holton-McKay duals as its witness Even Level Graph
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in a radial-by-level layout (source at centre, level-k vertices on ring k),
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coloured by parity, with the bridge switches highlighted: removed edges in
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red (dashed), added edges in green. Reads experiments/witnesses/dual_*.json.
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"""Draw the six Holton-McKay duals and their witness Even Level Graphs as
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crossing-free planar drawings (networkx planar_layout, Chrobak-Payne).
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Two figures:
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n21_elgs.png -- the six witness Even Level Graphs, parity-coloured,
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with the edges flipped by the bridge switches dashed red;
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n21_duals.png -- the six resulting duals, parity-coloured (same fixed
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parity labelling), with the introduced bridge edges green.
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Reads experiments/witnesses/dual_*.json. ELG and dual are drawn with
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independent planar layouts so neither has any edge crossing.
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"""
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import sys
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import os
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import json
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import math
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sys.path.insert(0, '/Users/didericis/Code/math-research/papers/'
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'level_resolutions_of_maximal_planar_graphs/experiments')
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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@@ -15,98 +19,95 @@ import matplotlib
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matplotlib.use('Agg')
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import matplotlib.pyplot as plt
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from matplotlib.lines import Line2D
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from test_conjecture import bfs_levels
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HERE = os.path.dirname(os.path.abspath(__file__))
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WDIR = os.path.join(HERE, 'witnesses')
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FDIR = os.path.join(HERE, '..', 'figures')
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EVEN_C = '#9ecae1' # even-level vertices
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ODD_C = '#fdae6b' # odd-level vertices
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EVEN_C = '#9ecae1' # even-parity vertices
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ODD_C = '#fdae6b' # odd-parity vertices
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def radial_pos(G, source):
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levels = bfs_levels(G, frozenset({source}))
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by_lvl = {}
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for v, k in levels.items():
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by_lvl.setdefault(k, []).append(v)
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pos = {}
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for k, verts in by_lvl.items():
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verts = sorted(verts)
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if k == 0:
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pos[verts[0]] = (0.0, 0.0)
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continue
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m = len(verts)
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for j, v in enumerate(verts):
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ang = 2 * math.pi * j / m + (k * 0.6)
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pos[v] = (k * math.cos(ang), k * math.sin(ang))
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return pos, levels
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def load(i):
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return json.load(open(os.path.join(WDIR, f'dual_{i}.json')))
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def draw(dual_index, ax):
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data = json.load(open(os.path.join(WDIR, f'dual_{dual_index}.json')))
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src = data['elg_source']
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def graph_of(edges, labels):
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G = nx.Graph()
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G.add_nodes_from(int(v) for v in data['labels'])
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G.add_edges_from((a, b) for a, b in data['elg_edges'])
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pos, levels = radial_pos(G, src)
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colors = [EVEN_C if levels[v] % 2 == 0 else ODD_C for v in G.nodes()]
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G.add_nodes_from(int(v) for v in labels)
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G.add_edges_from((a, b) for a, b in edges)
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return G
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removed = {frozenset(s['remove']) for s in data['bridge_switches']}
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added = [tuple(s['add']) for s in data['bridge_switches']]
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plain = [e for e in G.edges() if frozenset(e) not in removed]
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def draw(ax, G, labels, highlight, hcolor, hstyle, title):
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pos = nx.planar_layout(G)
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colors = [EVEN_C if labels[str(v)] == 0 else ODD_C for v in G.nodes()]
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hl = {frozenset(e) for e in highlight}
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plain = [e for e in G.edges() if frozenset(e) not in hl]
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nx.draw_networkx_edges(G, pos, edgelist=plain, ax=ax,
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edge_color='#bdbdbd', width=0.8)
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if removed:
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nx.draw_networkx_edges(G, pos, edgelist=[tuple(e) for e in removed],
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ax=ax, edge_color='#d62728', width=2.2,
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style='dashed')
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if added:
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nx.draw_networkx_edges(nx.Graph(added), pos, edgelist=added, ax=ax,
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edge_color='#2ca02c', width=2.2)
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nx.draw_networkx_nodes(G, pos, node_color=colors, node_size=210,
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edge_color='#b0b0b0', width=0.8)
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if hl:
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nx.draw_networkx_edges(G, pos, edgelist=[tuple(e) for e in hl], ax=ax,
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edge_color=hcolor, width=2.4, style=hstyle)
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nx.draw_networkx_nodes(G, pos, node_color=colors, node_size=200,
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edgecolors='#444444', linewidths=0.6, ax=ax)
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nx.draw_networkx_labels(G, pos, font_size=7, ax=ax)
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k = data['num_bridge_switches']
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title = (f'dual {dual_index}: ELG (source {src})'
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+ (f'\n{k} bridge switch' + ('es' if k != 1 else '')
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if k else '\n(Even Level Graph outright)'))
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ax.set_title(title, fontsize=9)
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ax.set_aspect('equal')
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nx.draw_networkx_labels(G, pos, font_size=8, ax=ax)
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ax.set_title(title, fontsize=10)
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ax.margins(0.12)
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ax.axis('off')
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def legend(fig, kind):
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handles = [
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Line2D([0], [0], marker='o', color='w', markerfacecolor=EVEN_C,
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markeredgecolor='#444', markersize=9, label='even parity'),
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Line2D([0], [0], marker='o', color='w', markerfacecolor=ODD_C,
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markeredgecolor='#444', markersize=9, label='odd parity'),
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]
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if kind == 'elg':
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handles.append(Line2D([0], [0], color='#d62728', lw=2.4, ls='dashed',
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label='edge flipped by a bridge switch'))
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else:
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handles.append(Line2D([0], [0], color='#2ca02c', lw=2.4,
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label='bridge edge introduced'))
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fig.legend(handles=handles, loc='lower center', ncol=3, fontsize=9,
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frameon=False)
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def main():
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os.makedirs(FDIR, exist_ok=True)
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# one combined 2x3 figure, plus individual files
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fig, axes = plt.subplots(2, 3, figsize=(13, 9))
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for i, ax in zip(range(6), axes.flat):
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draw(i, ax)
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legend = [
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Line2D([0], [0], marker='o', color='w', markerfacecolor=EVEN_C,
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markeredgecolor='#444', markersize=9, label='even level'),
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Line2D([0], [0], marker='o', color='w', markerfacecolor=ODD_C,
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markeredgecolor='#444', markersize=9, label='odd level'),
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Line2D([0], [0], color='#d62728', lw=2.2, ls='dashed',
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label='removed (flipped) edge'),
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Line2D([0], [0], color='#2ca02c', lw=2.2, label='added (bridge) edge'),
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]
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fig.legend(handles=legend, loc='lower center', ncol=4, fontsize=9,
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frameon=False)
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fig.tight_layout(rect=(0, 0.04, 1, 1))
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out = os.path.join(FDIR, 'n21_witnesses.png')
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fig.savefig(out, dpi=160)
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print(f'wrote {out}')
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for i in range(6):
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f, a = plt.subplots(figsize=(5, 5))
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draw(i, a)
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f.tight_layout()
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p = os.path.join(FDIR, f'n21_dual_{i}.png')
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f.savefig(p, dpi=160)
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plt.close(f)
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print(f'wrote {p}')
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# Figure 1: the six witness Even Level Graphs (flipped edges red dashed)
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fig, axes = plt.subplots(2, 3, figsize=(19, 12))
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for i, ax in zip(range(6), axes.flat):
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d = load(i)
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G = graph_of(d['elg_edges'], d['labels'])
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removed = [s['remove'] for s in d['bridge_switches']]
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k = d['num_bridge_switches']
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sub = (f"{k} bridge switch" + ("es" if k != 1 else "")
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if k else "Even Level Graph outright")
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draw(ax, G, d['labels'], removed, '#d62728', 'dashed',
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f"dual {i}: ELG (source {d['elg_source']})\n{sub}")
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legend(fig, 'elg')
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fig.tight_layout(rect=(0, 0.04, 1, 1))
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p1 = os.path.join(FDIR, 'n21_elgs.png')
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fig.savefig(p1, dpi=160)
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plt.close(fig)
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print(f'wrote {p1}')
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# Figure 2: the six resulting duals (introduced bridge edges green)
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fig, axes = plt.subplots(2, 3, figsize=(19, 12))
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for i, ax in zip(range(6), axes.flat):
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d = load(i)
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G = graph_of(d['dual_edges'], d['labels'])
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added = [s['add'] for s in d['bridge_switches']]
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draw(ax, G, d['labels'], added, '#2ca02c', 'solid', f"dual {i}")
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legend(fig, 'dual')
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fig.tight_layout(rect=(0, 0.04, 1, 1))
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p2 = os.path.join(FDIR, 'n21_duals.png')
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fig.savefig(p2, dpi=160)
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plt.close(fig)
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print(f'wrote {p2}')
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if __name__ == '__main__':
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@@ -49,5 +49,11 @@
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{6}{table.1}\protected@file@percent }
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\newlabel{tab:n21}{{1}{6}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.1}{}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{section*.3}\protected@file@percent }
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\gdef \@abspage@last{6}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The witness Even Level Graph for each of the six Holton--McKay duals, drawn as a crossing-free planar graph and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The dashed red edges are the same-parity edges that the bridge switches flip; flipping them yields the corresponding dual in Figure\nonbreakingspace \ref {fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs outright, so no edge is flipped.}}{7}{figure.5}\protected@file@percent }
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\newlabel{fig:n21-elgs}{{5}{7}{The witness Even Level Graph for each of the six Holton--McKay duals, drawn as a crossing-free planar graph and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The dashed red edges are the same-parity edges that the bridge switches flip; flipping them yields the corresponding dual in Figure~\ref {fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs outright, so no edge is flipped}{figure.5}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs with the same parity colouring. The solid green edges are the bridge edges introduced by the switches from the Even Level Graphs of Figure\nonbreakingspace \ref {fig:n21-elgs}. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{8}{figure.6}\protected@file@percent }
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\newlabel{fig:n21-duals}{{6}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs with the same parity colouring. The solid green edges are the bridge edges introduced by the switches from the Even Level Graphs of Figure~\ref {fig:n21-elgs}. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.6}{}}
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\gdef \@abspage@last{8}
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@@ -1,6 +1,6 @@
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# Fdb version 3
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["pdflatex"] 1779463291 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779463293
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(generated)
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"paper.log"
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@@ -566,6 +566,11 @@ INPUT ./fig_parity_subgraph.png
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INPUT fig_parity_subgraph.png
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INPUT ./fig_parity_subgraph.png
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INPUT ./fig_parity_subgraph.png
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INPUT ./figures/n21_witnesses.png
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INPUT ./figures/n21_witnesses.png
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INPUT figures/n21_witnesses.png
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INPUT ./figures/n21_witnesses.png
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INPUT ./figures/n21_witnesses.png
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INPUT paper.aux
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ve/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
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Output written on paper.pdf (6 pages, 550793 bytes).
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Output written on paper.pdf (8 pages, 1620175 bytes).
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PDF statistics:
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173 PDF objects out of 1000 (max. 8388607)
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129 compressed objects within 2 object streams
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33 named destinations out of 1000 (max. 500000)
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77 words of extra memory for PDF output out of 10000 (max. 10000000)
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196 PDF objects out of 1000 (max. 8388607)
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146 compressed objects within 2 object streams
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38 named destinations out of 1000 (max. 500000)
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87 words of extra memory for PDF output out of 10000 (max. 10000000)
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Binary file not shown.
@@ -428,15 +428,28 @@ witnesses are step-verified.}
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\begin{figure}[ht]
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\centering
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\includegraphics[width=\textwidth]{figures/n21_witnesses.png}
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\includegraphics[width=\textwidth]{figures/n21_elgs.png}
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\caption{The witness Even Level Graph for each of the six Holton--McKay
|
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duals, drawn radially by level (source at the centre, level-$k$ vertices on
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ring $k$) and coloured by parity (blue even, orange odd). Dashed red edges
|
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are the same-parity edges flipped by the bridge switches; solid green edges
|
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are the bridge edges they introduce; applying the switches turns each Even
|
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Level Graph into the corresponding dual. Duals $1$ and $2$ are Even Level
|
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Graphs outright, so no switch is shown.}
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\label{fig:n21-witnesses}
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duals, drawn as a crossing-free planar graph and coloured by parity (blue
|
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even, orange odd, with respect to the fixed level-parity labelling). The
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dashed red edges are the same-parity edges that the bridge switches flip;
|
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flipping them yields the corresponding dual in
|
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Figure~\ref{fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs
|
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outright, so no edge is flipped.}
|
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\label{fig:n21-elgs}
|
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\end{figure}
|
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\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{figures/n21_duals.png}
|
||||
\caption{The six Holton--McKay duals, drawn as crossing-free planar graphs
|
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with the same parity colouring. The solid green edges are the bridge edges
|
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introduced by the switches from the Even Level Graphs of
|
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Figure~\ref{fig:n21-elgs}. Each green edge is a bridge of its parity
|
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subgraph, so no new cycle -- and in particular no odd cycle -- is created;
|
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duals $1$ and $2$ coincide with their Even Level Graphs and have no added
|
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edge.}
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\label{fig:n21-duals}
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\end{figure}
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\begin{thebibliography}{9}
|
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|
||||
Reference in New Issue
Block a user