diff --git a/papers/coloring_nested_tire_graphs/notes/fig_theta.png b/papers/coloring_nested_tire_graphs/notes/fig_theta.png deleted file mode 100644 index af9cc38..0000000 Binary files a/papers/coloring_nested_tire_graphs/notes/fig_theta.png and /dev/null differ diff --git a/papers/coloring_nested_tire_graphs/notes/generate_figures.py b/papers/coloring_nested_tire_graphs/notes/generate_figures.py index 57c5a03..e4ecde6 100644 --- a/papers/coloring_nested_tire_graphs/notes/generate_figures.py +++ b/papers/coloring_nested_tire_graphs/notes/generate_figures.py @@ -10,8 +10,6 @@ Figures produced: fig_star.png -- star K_{1,3} with a proper 3-edge-coloring fig_blocktree.png -- a subcubic outerplanar example showing two 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 os @@ -245,30 +243,6 @@ def fig_blocktree(filename): 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(): out_dir = HERE 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_star(os.path.join(out_dir, 'fig_star.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}") diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.log b/papers/coloring_nested_tire_graphs/notes/menagerie.log index 98c8d48..52093b4 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.log +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.log @@ -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 restricted \write18 enabled. %&-line parsing enabled. @@ -295,43 +295,37 @@ File: fig_blocktree.png Graphic file (type png) Package pdftex.def Info: fig_blocktree.png used on input line 158. (pdftex.def) Requested size: 258.36668pt x 157.34296pt. - [3] - -File: fig_theta.png Graphic file (type 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) ) + [3] [4 <./fig_blocktree.png>] (./menagerie.aux) ) Here is how much of TeX's memory you used: - 4643 strings out of 478268 - 75478 string characters out of 5846347 + 4627 strings out of 478268 + 75226 string characters out of 5846347 373157 words of memory out of 5000000 - 22821 multiletter control sequences out of 15000+600000 - 481374 words of font info for 75 fonts, out of 8000000 for 9000 + 22806 multiletter control sequences out of 15000+600000 + 479481 words of font info for 67 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 55i,6n,63p,245b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s -{/usr/local/tex -live/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc} -Output written on menagerie.pdf (4 pages, 236404 bytes). +{/usr/local/texlive/2022/texm +f-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc} +Output written on menagerie.pdf (4 pages, 227201 bytes). PDF statistics: - 111 PDF objects out of 1000 (max. 8388607) + 109 PDF objects out of 1000 (max. 8388607) 60 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 31 words of extra memory for PDF output out of 10000 (max. 10000000) + 26 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.pdf b/papers/coloring_nested_tire_graphs/notes/menagerie.pdf index 549fe8e..6788494 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/menagerie.pdf and b/papers/coloring_nested_tire_graphs/notes/menagerie.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.tex b/papers/coloring_nested_tire_graphs/notes/menagerie.tex index f9482e4..3bf9e30 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.tex +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.tex @@ -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 $3$, 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}