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coloring_nested_tire_graphs: remove the 'Outside the menagerie' section

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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>
This commit is contained in:
didericis
2026-05-25 21:22:04 -04:00
parent e47f89918a
commit faecfb1a3a
5 changed files with 27 additions and 78 deletions
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papers/coloring_nested_tire_graphs/notes/fig_theta.png
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papers/coloring_nested_tire_graphs/notes/generate_figures.py
-27
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@@ -10,8 +10,6 @@ Figures produced:
fig_star.png -- star K_{1,3} with a proper 3-edge-coloring fig_star.png -- star K_{1,3} with a proper 3-edge-coloring
fig_blocktree.png -- a subcubic outerplanar example showing two fig_blocktree.png -- a subcubic outerplanar example showing two
cycle-blocks attached at a cut-vertex cycle-blocks attached at a cut-vertex
fig_theta.png -- theta(2,2,2) = K_{2,3}: NOT outerplanar (counter-
example, drawn in red)
""" """
import math import math
import os import os
@@ -245,30 +243,6 @@ def fig_blocktree(filename):
plt.close() plt.close()
def fig_theta(filename):
"""Theta graph theta(2,2,2) = K_{2,3}: outside the menagerie because
not outerplanar (contains K_{2,3} = itself). Drawn in red as a
counterexample."""
pos = {
0: (-1.0, 0.0),
1: (1.0, 0.0),
2: (0.0, 0.8),
3: (0.0, 0.0),
4: (0.0, -0.8),
}
edges = [(0, 2), (2, 1),
(0, 3), (3, 1),
(0, 4), (4, 1)]
# Color in muted gray-red, no specific edge coloring
ec = ['#aa4444'] * len(edges)
fig, ax = plt.subplots(figsize=(3.4, 2.4))
draw_graph(ax, pos, edges, edge_colors=ec, vertex_color='#aa4444')
ax.set_xlim(-1.5, 1.5); ax.set_ylim(-1.3, 1.3)
ax.set_aspect('equal'); ax.axis('off')
plt.savefig(filename, dpi=160, bbox_inches='tight')
plt.close()
def main(): def main():
out_dir = HERE out_dir = HERE
fig_path(os.path.join(out_dir, 'fig_path.png'), n=5) fig_path(os.path.join(out_dir, 'fig_path.png'), n=5)
@@ -276,7 +250,6 @@ def main():
fig_corona(os.path.join(out_dir, 'fig_corona.png'), n=5) fig_corona(os.path.join(out_dir, 'fig_corona.png'), n=5)
fig_star(os.path.join(out_dir, 'fig_star.png')) fig_star(os.path.join(out_dir, 'fig_star.png'))
fig_blocktree(os.path.join(out_dir, 'fig_blocktree.png')) fig_blocktree(os.path.join(out_dir, 'fig_blocktree.png'))
fig_theta(os.path.join(out_dir, 'fig_theta.png'))
print(f"Wrote figures to {out_dir}") print(f"Wrote figures to {out_dir}")
papers/coloring_nested_tire_graphs/notes/menagerie.log
+27 -33
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@@ -1,4 +1,4 @@
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 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
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -295,43 +295,37 @@ File: fig_blocktree.png Graphic file (type png)
<use fig_blocktree.png> <use fig_blocktree.png>
Package pdftex.def Info: fig_blocktree.png used on input line 158. Package pdftex.def Info: fig_blocktree.png used on input line 158.
(pdftex.def) Requested size: 258.36668pt x 157.34296pt. (pdftex.def) Requested size: 258.36668pt x 157.34296pt.
[3] [3] [4 <./fig_blocktree.png>] (./menagerie.aux) )
<fig_theta.png, id=38, 168.47943pt x 147.70181pt>
File: fig_theta.png Graphic file (type png)
<use fig_theta.png>
Package pdftex.def Info: fig_theta.png used on input line 183.
(pdftex.def) Requested size: 150.32504pt x 131.79225pt.
[4 <./fig_blocktree.png> <./fig_theta.png>] (./menagerie.aux) )
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papers/coloring_nested_tire_graphs/notes/menagerie.pdf
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papers/coloring_nested_tire_graphs/notes/menagerie.tex
-18
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@@ -176,23 +176,5 @@ cut-vertex constraint. For each cycle-block $B = C_n$ contributing
$2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing $2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing
$3$, this product is computable in time linear in $|V(G)|+|E(G)|$. $3$, this product is computable in time linear in $|V(G)|+|E(G)|$.
\section*{Outside the menagerie: $\theta(p,q,r)$ with all paths
$\ge 2$, i.e.\ $K_{2,3}$ subdivisions}
\begin{center}
\includegraphics[width=0.32\textwidth]{fig_theta.png}
\end{center}
The genuine theta graph $\theta(2,2,2) = K_{2,3}$ (and more generally any
$\theta(p,q,r)$ with $p,q,r\ge 2$) is \emph{not} outerplanar: $K_{2,3}$
is a forbidden minor for outerplanarity, and every $\theta(p,q,r)$ with
$\min(p,q,r)\ge 2$ contains $K_{2,3}$ as a topological minor. Such graphs
do \emph{not} arise as the interior dual subgraph of a partial tire dual
$D(T)$ in this paper: when the inner outerplanar graph $O$ has a bridge,
the interior dual subgraph is the polygon-with-one-chord case
$\theta(1, p, q)$ of \S6, where the path of length~$1$ is precisely the
dual of the bridge. In particular $D(T)$ \emph{is} subcubic outerplanar
in the bridge case, and its edge $3$-coloring count is the
$\theta(1,p,q)$ closed form above.
\end{document} \end{document}