coloring_nested_tire_graphs: remove the 'Outside the menagerie' section
θ(1, p, q) IS in the menagerie (= subcubic outerplanar), so the section about θ being outside no longer applies. Removed: - menagerie.tex: the entire 'Outside the menagerie' section. - generate_figures.py: fig_theta function and its main() call. - fig_theta.png: orphaned figure file. Note: the paper now ends cleanly with the block-cut decomposition section. 4 pages. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -10,8 +10,6 @@ Figures produced:
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fig_star.png -- star K_{1,3} with a proper 3-edge-coloring
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fig_blocktree.png -- a subcubic outerplanar example showing two
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cycle-blocks attached at a cut-vertex
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fig_theta.png -- theta(2,2,2) = K_{2,3}: NOT outerplanar (counter-
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example, drawn in red)
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"""
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import math
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import os
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@@ -245,30 +243,6 @@ def fig_blocktree(filename):
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plt.close()
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def fig_theta(filename):
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"""Theta graph theta(2,2,2) = K_{2,3}: outside the menagerie because
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not outerplanar (contains K_{2,3} = itself). Drawn in red as a
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counterexample."""
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pos = {
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0: (-1.0, 0.0),
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1: (1.0, 0.0),
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2: (0.0, 0.8),
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3: (0.0, 0.0),
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4: (0.0, -0.8),
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}
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edges = [(0, 2), (2, 1),
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(0, 3), (3, 1),
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(0, 4), (4, 1)]
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# Color in muted gray-red, no specific edge coloring
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ec = ['#aa4444'] * len(edges)
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fig, ax = plt.subplots(figsize=(3.4, 2.4))
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draw_graph(ax, pos, edges, edge_colors=ec, vertex_color='#aa4444')
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ax.set_xlim(-1.5, 1.5); ax.set_ylim(-1.3, 1.3)
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ax.set_aspect('equal'); ax.axis('off')
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plt.savefig(filename, dpi=160, bbox_inches='tight')
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plt.close()
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def main():
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out_dir = HERE
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fig_path(os.path.join(out_dir, 'fig_path.png'), n=5)
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@@ -276,7 +250,6 @@ def main():
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fig_corona(os.path.join(out_dir, 'fig_corona.png'), n=5)
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fig_star(os.path.join(out_dir, 'fig_star.png'))
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fig_blocktree(os.path.join(out_dir, 'fig_blocktree.png'))
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fig_theta(os.path.join(out_dir, 'fig_theta.png'))
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print(f"Wrote figures to {out_dir}")
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@@ -1,4 +1,4 @@
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:18
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:21
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entering extended mode
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restricted \write18 enabled.
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%&-line parsing enabled.
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@@ -295,43 +295,37 @@ File: fig_blocktree.png Graphic file (type png)
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<use fig_blocktree.png>
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Package pdftex.def Info: fig_blocktree.png used on input line 158.
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(pdftex.def) Requested size: 258.36668pt x 157.34296pt.
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[3]
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<fig_theta.png, id=38, 168.47943pt x 147.70181pt>
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File: fig_theta.png Graphic file (type png)
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<use fig_theta.png>
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Package pdftex.def Info: fig_theta.png used on input line 183.
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(pdftex.def) Requested size: 150.32504pt x 131.79225pt.
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[4 <./fig_blocktree.png> <./fig_theta.png>] (./menagerie.aux) )
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[3] [4 <./fig_blocktree.png>] (./menagerie.aux) )
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55i,6n,63p,245b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s
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{/usr/local/tex
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live/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc}</usr/local/texl
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Output written on menagerie.pdf (4 pages, 236404 bytes).
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f-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf
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Output written on menagerie.pdf (4 pages, 227201 bytes).
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PDF statistics:
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Binary file not shown.
@@ -176,23 +176,5 @@ cut-vertex constraint. For each cycle-block $B = C_n$ contributing
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$2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing
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$3$, this product is computable in time linear in $|V(G)|+|E(G)|$.
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\section*{Outside the menagerie: $\theta(p,q,r)$ with all paths
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$\ge 2$, i.e.\ $K_{2,3}$ subdivisions}
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\begin{center}
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\includegraphics[width=0.32\textwidth]{fig_theta.png}
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\end{center}
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The genuine theta graph $\theta(2,2,2) = K_{2,3}$ (and more generally any
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$\theta(p,q,r)$ with $p,q,r\ge 2$) is \emph{not} outerplanar: $K_{2,3}$
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is a forbidden minor for outerplanarity, and every $\theta(p,q,r)$ with
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$\min(p,q,r)\ge 2$ contains $K_{2,3}$ as a topological minor. Such graphs
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do \emph{not} arise as the interior dual subgraph of a partial tire dual
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$D(T)$ in this paper: when the inner outerplanar graph $O$ has a bridge,
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the interior dual subgraph is the polygon-with-one-chord case
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$\theta(1, p, q)$ of \S6, where the path of length~$1$ is precisely the
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dual of the bridge. In particular $D(T)$ \emph{is} subcubic outerplanar
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in the bridge case, and its edge $3$-coloring count is the
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$\theta(1,p,q)$ closed form above.
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\end{document}
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