diff --git a/papers/coloring_nested_tire_graphs/experiments/tire_def_figure.py b/papers/coloring_nested_tire_graphs/experiments/tire_def_figure.py new file mode 100644 index 0000000..1e95d44 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/tire_def_figure.py @@ -0,0 +1,113 @@ +"""Generate the example figure for Definition 1.5 (tire graph) of the +paper. Produces fig_tire_example.png at the paper's top level. + +Picks a specific small tire (m=6 outer cycle, k=4 inner cycle with one +chord) so that all four named parts B_out, O, B_in, E_ann are visible +and the annular triangles are individually legible. +""" +import math +import os +import sys + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +import matplotlib.pyplot as plt +import matplotlib.patches as patches + +from tire_graph import random_tire, planar_positions + + +def draw_tire_def(tire, filename): + m, k = tire['m'], tire['k'] + outer, inner = tire['outer'], tire['inner'] + edges = tire['edges'] + + R_out, R_in = 1.0, 0.45 + pos = planar_positions(tire, R_out=R_out, R_in=R_in) + + fig, ax = plt.subplots(figsize=(6.5, 6.5)) + + # guide circles for reference + for r in (R_out + 0.04, R_in - 0.04): + ax.add_patch(patches.Circle((0, 0), r, fill=False, + edgecolor='lightgray', + linewidth=0.5, linestyle='--')) + + outer_set = set(outer) + inner_set = set(inner) + C = { + 'outer_cycle': '#1f77b4', + 'inner_cycle': '#d62728', + 'inner_chord': '#ff7f0e', + 'spoke': '#7f7f7f', + } + + # classify and draw edges + for (u, v) in edges: + if u in outer_set and v in outer_set: + color, lw, label = C['outer_cycle'], 2.6, 'B_out' + elif u in inner_set and v in inner_set: + ia, ib = u - m, v - m + d = abs(ia - ib) + d = min(d, k - d) + if d == 1: + color, lw, label = C['inner_cycle'], 2.6, 'B_in' + else: + color, lw, label = C['inner_chord'], 1.8, 'O chord' + else: + color, lw, label = C['spoke'], 1.1, 'E_ann' + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1) + + # vertices + for v in outer: + x, y = pos[v] + ax.plot(x, y, 'o', color=C['outer_cycle'], markersize=14, zorder=2) + ax.annotate(str(v), (x, y), color='white', ha='center', + va='center', fontsize=9, fontweight='bold', zorder=3) + for v in inner: + x, y = pos[v] + ax.plot(x, y, 'o', color=C['inner_cycle'], markersize=13, zorder=2) + ax.annotate(str(v), (x, y), color='white', ha='center', + va='center', fontsize=8, fontweight='bold', zorder=3) + + # legend + legend_items = [ + plt.Line2D([], [], color=C['outer_cycle'], linewidth=2.6, + label=r'$B_{\mathrm{out}}$ (outer boundary, $m=6$)'), + plt.Line2D([], [], color=C['inner_cycle'], linewidth=2.6, + label=r'$B_{\mathrm{in}}$ (inner boundary, $k=4$)'), + plt.Line2D([], [], color=C['inner_chord'], linewidth=1.8, + label=r'chord of $O$'), + plt.Line2D([], [], color=C['spoke'], linewidth=1.1, + label=r'$E_{\mathrm{ann}}$ (annular edges)'), + ] + ax.legend(handles=legend_items, loc='upper left', + bbox_to_anchor=(1.0, 1.0), fontsize=10, frameon=False) + + ax.set_xlim(-1.20, 1.20) + ax.set_ylim(-1.20, 1.20) + ax.set_aspect('equal') + ax.axis('off') + + plt.savefig(filename, dpi=160, bbox_inches='tight') + plt.close() + + +def main(): + # m=6 outer, k=4 inner, 1 chord — try a few seeds to pick a clean one + paper_dir = os.path.abspath(os.path.join(HERE, '..')) + candidates = [(6, 4, 1, s) for s in (3, 5, 8, 13, 21)] + # Use seed=3 as the chosen example (good lattice-path balance) + for (m, k, nc, seed) in candidates[:1]: + tire = random_tire(m, k, n_chords=nc, seed=seed) + fn = os.path.join(paper_dir, 'fig_tire_example.png') + draw_tire_def(tire, fn) + print(f" wrote {fn}") + print(f" m={m}, k={k}, chords={tire['inner_chords']}, " + f"path={tire['lattice_path']}") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/fig_tire_example.png b/papers/coloring_nested_tire_graphs/fig_tire_example.png new file mode 100644 index 0000000..c4f4316 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_tire_example.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 450c2a8..44ee562 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -1,12 +1,17 @@ \relax \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \newlabel{def:dual-depth}{{1.4}{1}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } +\newlabel{fig:dual-depth}{{1}{2}} +\newlabel{def:tire-graph}{{1.5}{2}} +\newlabel{rem:tire-counts}{{1.6}{2}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } -\newlabel{fig:dual-depth}{{1}{2}} -\newlabel{def:tire-graph}{{1.5}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } +\newlabel{fig:tire-example}{{2}{3}} +\newlabel{lem:tire-component}{{1.7}{3}} +\newlabel{rem:tire-component-degenerate}{{1.8}{3}} \gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index 004a41f..0923669 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779733650 "paper.tex" "paper.pdf" "paper" 1779733650 +["pdflatex"] 1779735598 "paper.tex" "paper.pdf" "paper" 1779735599 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -21,7 +21,9 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" @@ -56,8 +58,9 @@ "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "fig_dual_depth.png" 1779482522 255786 cb48aab5aa40fc161d13a75df0544511 "" - 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PDF statistics: - 77 PDF objects out of 1000 (max. 8388607) - 45 compressed objects within 1 object stream + 89 PDF objects out of 1000 (max. 8388607) + 51 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 6 words of extra memory for PDF output out of 10000 (max. 10000000) + 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 5a7f442..ef963f7 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 3875fca..f05d45d 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -116,39 +116,94 @@ vertex.} \begin{definition}[Tire graph] \label{def:tire-graph} -Let $C_{\mathrm{out}}$ be a simple cycle of length $m \geq 3$, and let -$O$ be an outerplanar graph whose outer-face boundary $C_{\mathrm{in}}$ -is a simple cycle of length $k \geq 3$, with $V(C_{\mathrm{out}}) \cap -V(O) = \emptyset$. A \emph{tire graph} on $(C_{\mathrm{out}}, O)$ is a -plane graph $T$ with +A \emph{tire graph} consists of a plane graph $T$ together with two +\emph{boundary parts} $B_{\mathrm{out}}, B_{\mathrm{in}} \subseteq T$ +and an \emph{inner outerplanar graph} $O \subseteq T$, where each of +$B_{\mathrm{out}}$ and the outer-face boundary $B_{\mathrm{in}}$ of $O$ +is either +\begin{itemize} + \item a simple cycle of length $\geq 3$, or + \item a single vertex (a \emph{degenerate} boundary), +\end{itemize} +with at most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ degenerate, and +$V(B_{\mathrm{out}}) \cap V(O) = \emptyset$. The vertex and edge sets +of $T$ are \[ - V(T) = V(C_{\mathrm{out}}) \cup V(O), + V(T) = V(B_{\mathrm{out}}) \cup V(O), \qquad - E(T) = E(C_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, + E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, \] -where $E_{\mathrm{ann}}$ is a set of edges --- the \emph{annular edges} ---- such that, in the plane embedding of $T$, the closed annulus with -outer boundary $C_{\mathrm{out}}$ and inner boundary $C_{\mathrm{in}}$ -is partitioned into triangular faces. Equivalently, the bounded faces -of $T$ that are not faces of $O$ are all triangles, and together they -tile the annular region between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$. +where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the +property that, in the plane embedding of $T$, the closed planar region +$R$ bounded externally by $B_{\mathrm{out}}$ and internally by +$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose +union is $R$. The region $R$ is a closed annulus when both +$B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ are cycles, and a closed disk +when exactly one of them is a single vertex. -We call $C_{\mathrm{out}}$ the \emph{outer cycle}, $O$ the \emph{inner -outerplanar graph}, and $C_{\mathrm{in}}$ the \emph{inner cycle} of -$T$. When $O = C_{\mathrm{in}}$ (the inner outerplanar graph has no -chords), $T$ is a tire graph \emph{with empty inner}; in general $O$ -contributes only chords inside the disk bounded by $C_{\mathrm{in}}$ -and does not interact with $E_{\mathrm{ann}}$. +We call $B_{\mathrm{out}}$ the \emph{outer boundary}, $O$ the +\emph{inner outerplanar graph}, and $B_{\mathrm{in}}$ the \emph{inner +boundary} of $T$. A tire graph in which $B_{\mathrm{out}}$ +(respectively $B_{\mathrm{in}}$) is a single vertex is said to have a +\emph{degenerate outer (respectively inner) boundary}; in either case +$T$ is a triangulated closed disk with that vertex as apex. \end{definition} +\begin{figure}[h] +\centering +\includegraphics[width=0.78\textwidth]{fig_tire_example.png} +\caption{A tire graph with non-degenerate boundaries: outer boundary +$B_{\mathrm{out}}$ a $6$-cycle on vertices $0,\dots,5$ (blue), inner +boundary $B_{\mathrm{in}}$ a $4$-cycle on vertices $6,\dots,9$ (red), +inner outerplanar graph $O = B_{\mathrm{in}} \cup \{7\text{--}9\}$ +(with one chord, orange), and $E_{\mathrm{ann}}$ (grey) tiling the +annulus between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ by ten +triangular faces.} +\label{fig:tire-example} +\end{figure} + \begin{remark} -A tire graph on $(C_{\mathrm{out}}, O)$ has $|V(C_{\mathrm{out}})| + -|V(O)| = m + k$ vertices, exactly $m + k$ annular triangles -in the annulus between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$ (by -Euler's formula on the annulus), and exactly $m + k$ annular edges -in $E_{\mathrm{ann}}$, of which the $m + k$ triangles share their -three edges with the boundaries $E(C_{\mathrm{out}}) \cup -E(C_{\mathrm{in}})$ and with each other. +\label{rem:tire-counts} +Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By +Euler's formula on the annular (resp.\ disk) region $R$, the tire graph +has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ +annular edges when neither boundary is degenerate; when exactly one +boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ +triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. +\end{remark} + +\begin{lemma}[Tire-component lemma] +\label{lem:tire-component} +Let $G$ be a maximal planar graph with fixed embedding $\Pi_G$ and let +$S \subseteq V(G)$ be a level source. For $d \geq 0$, let +\[ + G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr] +\] +be the inner-dual subgraph on dual vertices of dual depth $d$, and let +$C'$ be a connected component of $G'_d$. Write +$F_{C'} := \{f : d_f \in V(C')\}$, +$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and +$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$. + +Then $C$, together with its inherited embedding, is a tire graph in the +sense of Definition~\ref{def:tire-graph}: the two boundary parts +$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$ +subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph +$G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of +$C$ inside its closed boundary region are exactly the faces of $G$ in +$F_{C'}$. +\end{lemma} + +\begin{remark} +\label{rem:tire-component-degenerate} +Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be +degenerate. For instance, at $d = 0$ with single-vertex source +$S = \{v_0\}$ the unique component of $G'_0$ has +$V_{C'} \cap L_0 = \{v_0\}$ --- the degenerate boundary --- and +$V_{C'} \cap L_1$ a cycle (the link of $v_0$ in $G$). Which of the two +parts is $B_{\mathrm{out}}$ and which is $B_{\mathrm{in}}$ depends on +the orientation of the inherited embedding (equivalently, on which side +of $C$ contains the rest of $\Pi_G$). \end{remark} \end{document}