From 8faf37a9dc8417d2be4da0dfd9193cc0b75c745f Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 22:42:14 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add Figure 5 illustrating Def 1.15 in the bridge case MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds fig_facial_dual_choices.png showing T'_{f'} for the three faces A, B, C of T'_ann = θ(1, 3, 3) in the bridge case: Face A (outer): V(A) = all 6 vertices; T'_A pulls in all four external neighbors u_1, u_2, u_4, u_5. Face B (inner right): V(B) = {v_0, v_1, v_2, v_3}; T'_B pulls in only u_1, u_2 and the cycle-neighbors v_4, v_5. Face C (inner left): mirror image of B. The figure also visually conveys that the chord endpoints v_0, v_3 (= the two annular faces sharing the bridge edge) have all three G'-edges inside T'_ann, so neither contributes an external u_v. Adds: - experiments/draw_facial_dual_choices.py - notes/fig_facial_dual_choices.png - Figure 5 in paper.tex referencing the new image. Paper grows from 9 to 10 pages. Co-Authored-By: Claude Opus 4.7 --- .../experiments/draw_facial_dual_choices.py | 171 ++++++++++++++++++ .../notes/fig_facial_dual_choices.png | Bin 0 -> 75719 bytes papers/coloring_nested_tire_graphs/paper.aux | 4 +- papers/coloring_nested_tire_graphs/paper.log | 29 ++- papers/coloring_nested_tire_graphs/paper.pdf | Bin 752040 -> 814715 bytes papers/coloring_nested_tire_graphs/paper.tex | 20 ++ 6 files changed, 213 insertions(+), 11 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py new file mode 100644 index 0000000..844e44c --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py @@ -0,0 +1,171 @@ +"""For the bridge case where T'_ann = theta(1, p, q) has three faces, +illustrate how the partial tire facial dual T'_{f'} (Definition 1.15) +depends on the choice of face f'. + +We use theta(1, 3, 3) = C_6 + chord (v_0, v_3) as the smallest +example. In the inherited planar embedding theta(1, 3, 3) has three +faces: + + Face A (outer/unbounded): V(A) = {v_0, ..., v_5} (all 6 vertices). + Face B (inner, right side): V(B) = {v_0, v_1, v_2, v_3}. + Face C (inner, left side): V(C) = {v_0, v_3, v_4, v_5}. + +In the surrounding maximal-planar G, each non-chord annular dual +vertex (v_1, v_2, v_4, v_5) has one external G'-neighbor (corresponding +to a non-annular face of G across a boundary edge of T). The chord +endpoints v_0, v_3 have all three G'-edges inside T'_ann (= 2 cycle +edges + chord edge), so no external neighbors. + +T'_{f'} is the closed G'-neighborhood of V(f') together with every +G'-edge incident to V(f'). +""" +import math +import os +import matplotlib.pyplot as plt +import matplotlib.patches as patches + + +# Hexagon positions: v_0 at top, v_3 at bottom, so chord v_0-v_3 is vertical. +def hex_pos(scale=1.0): + pos = {} + for i in range(6): + angle = math.pi / 2 - 2 * math.pi * i / 6 + pos[i] = (scale * math.cos(angle), scale * math.sin(angle)) + return pos + + +# External neighbors of v_1, v_2, v_4, v_5 (drawn radially outward). +def ext_pos(scale=1.0, radial=1.8): + pos = hex_pos(scale) + ext = {} + for v in (1, 2, 4, 5): + x, y = pos[v] + nx, ny = x / math.hypot(x, y), y / math.hypot(x, y) + ext[v] = (radial * nx, radial * ny) + return ext + + +CYCLE_EDGES = [(i, (i + 1) % 6) for i in range(6)] +CHORD = (0, 3) + + +def draw_panel(ax, vfp, title, show_external_for, face_shade=None, + face_shade_color='#ffe9a1', external_labels=True): + """Render one panel showing T'_ann (faint) with V(f') highlighted + (dark) and T'_{f'} edges + external neighbors highlighted.""" + pos = hex_pos() + ext = ext_pos() + + # Shade the chosen face (optional) + if face_shade is not None: + ax.add_patch(patches.Polygon(face_shade, closed=True, + facecolor=face_shade_color, + edgecolor='none', alpha=0.55, zorder=0)) + + # Draw faint version of T'_ann (cycle + chord) underneath + for u, v in CYCLE_EDGES: + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color='#cccccc', linewidth=2.0, + zorder=1) + x1, y1 = pos[CHORD[0]]; x2, y2 = pos[CHORD[1]] + ax.plot([x1, x2], [y1, y2], color='#cccccc', linewidth=2.0, + linestyle=(0, (3, 2)), zorder=1) + + # Highlight T'_{f'} edges: any G'-edge incident to V(f') + highlight = '#1f77b4' + for u, v in CYCLE_EDGES: + if u in vfp or v in vfp: + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=highlight, linewidth=3.2, + zorder=2) + if CHORD[0] in vfp or CHORD[1] in vfp: + x1, y1 = pos[CHORD[0]]; x2, y2 = pos[CHORD[1]] + ax.plot([x1, x2], [y1, y2], color=highlight, linewidth=3.2, + linestyle=(0, (3, 2)), zorder=2) + # External edges from V(f') ∩ {v_1, v_2, v_4, v_5} + for v in show_external_for: + if v in vfp: + x1, y1 = pos[v]; x2, y2 = ext[v] + ax.plot([x1, x2], [y1, y2], color=highlight, linewidth=3.2, + zorder=2) + + # Draw vertices: V(f') dark, others light + for v in range(6): + x, y = pos[v] + if v in vfp: + ax.plot(x, y, 'o', color='#1f3f70', markersize=18, zorder=3) + tcol = 'white' + else: + ax.plot(x, y, 'o', color='#bbbbbb', markersize=14, zorder=3) + tcol = 'white' + ax.annotate(f"$v_{v}$", (x, y), color=tcol, ha='center', + va='center', fontsize=9, fontweight='bold', zorder=4) + + # External neighbors (drawn only if they're added to T'_{f'}) + for v in show_external_for: + if v in vfp: + x, y = ext[v] + ax.plot(x, y, 's', color='#d62728', markersize=12, zorder=3) + if external_labels: + ax.annotate(f"$u_{v}$", (x, y), color='white', ha='center', + va='center', fontsize=7, fontweight='bold', + zorder=4) + + ax.set_xlim(-2.2, 2.2) + ax.set_ylim(-2.2, 2.2) + ax.set_aspect('equal'); ax.axis('off') + ax.set_title(title, fontsize=10) + + +def main(): + here = os.path.dirname(os.path.abspath(__file__)) + out = os.path.join(here, '..', 'notes', 'fig_facial_dual_choices.png') + out = os.path.normpath(out) + + pos = hex_pos() + + fig, axes = plt.subplots(1, 3, figsize=(12.0, 4.6)) + + # External neighbors only exist for non-chord vertices + ext_candidates = {1, 2, 4, 5} + + # ---- Face A: outer (all 6 vertices) ---- + vA = {0, 1, 2, 3, 4, 5} + # Outer face shading: a frame outside the hexagon. Skip explicit shading. + draw_panel(axes[0], vA, + title=r"Face $A$ (outer): $V(A) = \{v_0,\dots,v_5\}$" + "\n" + + r"$T'_{f'=A}$: full $\theta(1,3,3)$ + 4 external neighbors", + show_external_for=ext_candidates) + + # ---- Face B: inner right (v_0, v_1, v_2, v_3) ---- + vB = {0, 1, 2, 3} + shadeB = [pos[i] for i in (0, 1, 2, 3)] + draw_panel(axes[1], vB, + title=r"Face $B$ (inner right): $V(B) = \{v_0,v_1,v_2,v_3\}$" + "\n" + + r"$T'_{f'=B}$: includes $v_4, v_5$ via $v_0v_5, v_3v_4$", + show_external_for=ext_candidates, + face_shade=shadeB) + + # ---- Face C: inner left (v_0, v_3, v_4, v_5) ---- + vC = {0, 3, 4, 5} + shadeC = [pos[i] for i in (0, 5, 4, 3)] + draw_panel(axes[2], vC, + title=r"Face $C$ (inner left): $V(C) = \{v_0,v_3,v_4,v_5\}$" + "\n" + + r"$T'_{f'=C}$: mirror image of $B$", + show_external_for=ext_candidates, + face_shade=shadeC) + + fig.suptitle(r"Partial tire facial dual $T'_{f'}$ for the bridge case " + + r"($T'_{\mathrm{ann}} = \theta(1,3,3)$, three faces $A,B,C$)" + "\n" + + r"Blue: edges of $T'_{f'}$. Dark circles: $V(f')$. 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v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{5}{10}} +\gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index ba18c94..45a1d1e 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.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 22:36 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 22:42 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -228,16 +228,25 @@ LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 575. -[9] (./paper.aux) + +File: notes/fig_facial_dual_choices.png Graphic file (type png) + +Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line +627. +(pdftex.def) Requested size: 360.0pt x 143.50418pt. + +LaTeX Warning: `h' float specifier changed to `ht'. + +[9] [10 <./notes/fig_facial_dual_choices.png>] (./paper.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: - 3031 strings out of 478268 - 42740 string characters out of 5846347 - 344259 words of memory out of 5000000 - 21076 multiletter control sequences out of 15000+600000 + 3040 strings out of 478268 + 43027 string characters out of 5846347 + 344270 words of memory out of 5000000 + 21083 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,8n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -257,10 +266,10 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (9 pages, 752040 bytes). +Output written on paper.pdf (10 pages, 814715 bytes). 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zD4Ur>Gf3H+nt2lHFcLB`v+6-Js91TM{l|o6&>+-dB4iaN|25~w z(f?@B6S8o!5EA}}HIf2AHr1`5VuU(!_dv6~8lN1;bc?C^Op%aA8wYWmr z9x%GYz^%R;-HXf_rw%rEXt9O0-w(CmenNXOO@Uv~_fMc??+>ljkR;H{Ol&NfGPzw5 z+A!~2IQJRh72{cLM^sW2W9SOEJ~5PYhAxp*BC&=P#Tl{Hyqvcr!x_ofA-3$RN}PVU zmWV-=Kufz diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 5d75edd..c407ce3 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -622,6 +622,26 @@ $B_{\mathrm{out}}$ and one $O$-side face inside $B_{\mathrm{in}}$, $T'_{f'}$ recovers the planar dual of $T$ itself. \end{remark} +\begin{figure}[h] +\centering +\includegraphics[width=\textwidth]{notes/fig_facial_dual_choices.png} +\caption{\label{fig:facial-dual-choices} +The bridge case: $T'_{\mathrm{ann}} = \theta(1, 3, 3)$ has three faces +$A, B, C$ in its inherited embedding, with respective vertex sets +$V(A) = \{v_0, \dots, v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and +$V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar +$G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing +the bridge edge) have all three $G'$-edges inside $T'_{\mathrm{ann}}$, +while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes +one $G'$-edge to an external non-annular neighbor $u_i$. Each panel +highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, +gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm{ann}}$, +and red squares are external $G'$-neighbors $u_i$. The choice of +face $f'$ controls which external neighbors $u_i$ are pulled into +$T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ +and face $C$ pulls in $u_4, u_5$).} +\end{figure} + \begin{thebibliography}{9} \bibitem{bauerfeld-pds}