From b4fece08bc7bad12658647d99e261ca2819f13a0 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 15:04:04 -0400 Subject: [PATCH] coloring_nested_tire_graphs: extend Def 1.5 with degenerate boundaries; add tire-component lemma + example figure MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Extends Definition 1.5 (tire graph) to allow either B_out or B_in to be a single vertex (a "degenerate" boundary); the tire is then a triangulated disk with that vertex as apex. - Adds Remark 1.6 with vertex/edge/triangle counts for both the non-degenerate (annulus) and degenerate (disk) cases. - Adds Lemma 1.7 (tire-component lemma): for a maximal planar graph G with level source S and depth d ≥ 0, every connected component C' of the dual subgraph on dual vertices of depth d induces a subgraph C on G that is a tire graph; its two boundary parts are the level-d and level-(d+1) induced subgraphs (in either order), and its triangular faces are exactly the faces of G in F_{C'}. - Adds Remark 1.8 noting the d=0 degenerate-source case as an example. - Adds a figure (fig_tire_example.png) illustrating the definition with m=6 outer cycle, k=4 inner cycle, one chord in O, plus a legend identifying B_out, B_in, O's chord, and E_ann. - Adds experiments/tire_def_figure.py to regenerate the figure. Co-Authored-By: Claude Opus 4.7 --- .../experiments/tire_def_figure.py | 113 ++++++++++++++++++ .../fig_tire_example.png | Bin 0 -> 104494 bytes papers/coloring_nested_tire_graphs/paper.aux | 11 +- .../paper.fdb_latexmk | 9 +- papers/coloring_nested_tire_graphs/paper.fls | 9 +- papers/coloring_nested_tire_graphs/paper.log | 59 +++++---- papers/coloring_nested_tire_graphs/paper.pdf | Bin 342003 -> 449333 bytes papers/coloring_nested_tire_graphs/paper.tex | 107 +++++++++++++---- 8 files changed, 251 insertions(+), 57 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/tire_def_figure.py create mode 100644 papers/coloring_nested_tire_graphs/fig_tire_example.png 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 0000000000000000000000000000000000000000..c4f431657a70b45bbd0cefbf5c0fe724134b7d7a GIT binary patch literal 104494 zcmeFZWmr^Q8$U`j(uj0}gd>fBbhk(j-67o}-3*`z2m;b2Fbv&YqEaf|-Q67nXY;(z zbI$+c`EtIUeO<_f!=62RueI*=tD9)GxAM4H6j(?|NVtj$G8#xoXp=}ts1hIy;D11z zAJu_xLLRcZ9-7YYJl>nRSs|&IdAK+@dpOvd(|K9Bx!XEB@p5o;aqzLx*?4%kxC?P| zI{x3k!Qt#?&8c^5BMiLD6Bh+NcO)dT^v5q`MbEG=NXSS?iZYT~-WhxIm>yc|6BtV> 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\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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