From e212ada649d3eafe50b95a1a58bb44686fb8b56d Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 18:37:19 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add figure of partial tire dual MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds fig_partial_tire_dual.png next to Definition 1.7 (Partial tire dual), illustrating the corona-graph structure C_{n+m} ∘ K_1 from Proposition 1.8 on a concrete m=6, k=4 spoke-only tire. Drawing details: - d_f interior vertices (purple squares) at centroids of annular triangles; the 10 of them form a 10-cycle (solid purple edges). - B_out leaves (orange diamonds) placed outside B_out; the leaf edge is dashed orange. - B_in leaves placed on the central-hole side of each red edge, at a fixed small offset past the midpoint of the edge in the d_f -> midpoint direction. This positions each leaf edge to cross its red edge exactly at the midpoint while keeping the leaf vertex itself inside the central hole and off the border. Generator: experiments/draw_partial_tire_dual.py (also committed). Co-Authored-By: Claude Opus 4.7 --- .../experiments/draw_partial_tire_dual.py | 221 ++++++++++++++++++ .../experiments/partial_tire_dual_example.png | Bin 0 -> 150413 bytes .../fig_partial_tire_dual.png | Bin 0 -> 150413 bytes papers/coloring_nested_tire_graphs/paper.aux | 10 +- papers/coloring_nested_tire_graphs/paper.log | 60 ++--- papers/coloring_nested_tire_graphs/paper.pdf | Bin 499770 -> 616266 bytes papers/coloring_nested_tire_graphs/paper.tex | 15 ++ 7 files changed, 275 insertions(+), 31 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual.py create mode 100644 papers/coloring_nested_tire_graphs/experiments/partial_tire_dual_example.png create mode 100644 papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual.py b/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual.py new file mode 100644 index 0000000..8de88ab --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual.py @@ -0,0 +1,221 @@ +"""Draw a partial tire dual on top of a small tire graph. + +Uses a spoke-only tire with m=6 outer cycle, k=4 inner cycle, no +chords. Overlays the partial tire dual: + - d_f vertices at centroids of annular triangles (purple squares). + - Leaf vertices for B_out edges (orange diamonds, outside the tire). + - Leaf vertices for B_in edges (orange diamonds, inside the inner hole). + - Interior dual edges (dashed purple). + - Leaf edges (dashed orange). +""" +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_partial_tire_dual(tire, filename): + m, k = tire['m'], tire['k'] + outer, inner = tire['outer'], tire['inner'] + triangles = tire['triangles'] + lattice_path = tire['lattice_path'] + + 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=(9, 9)) + + # === Draw underlying tire graph faintly === + outer_set = set(outer) + inner_set = set(inner) + for u, v in tire['edges']: + x1, y1 = pos[u]; x2, y2 = pos[v] + if u in outer_set and v in outer_set: + color = '#a8c9e8'; lw = 2.0 + elif u in inner_set and v in inner_set: + color = '#e8a8a8'; lw = 2.0 + else: + color = '#cccccc'; lw = 1.0 + ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1) + for v in outer: + x, y = pos[v] + ax.plot(x, y, 'o', color='#a8c9e8', markersize=14, zorder=2) + ax.annotate(str(v), (x, y), color='white', ha='center', va='center', + fontsize=8, fontweight='bold', zorder=3) + for v in inner: + x, y = pos[v] + ax.plot(x, y, 'o', color='#e8a8a8', markersize=12, zorder=2) + ax.annotate(str(v), (x, y), color='white', ha='center', va='center', + fontsize=7, fontweight='bold', zorder=3) + + # === Reconstruct annular triangles in cyclic order === + # The triangles list from random_tire gives them in lattice-path order. + print(f"Lattice path: {lattice_path}") + print(f"Triangles: {triangles}") + n_tri = len(triangles) + assert n_tri == m + k, f"expected {m+k} triangles, got {n_tri}" + + # Compute centroid of each triangle + centroids = [] + for tri in triangles: + v1, v2, v3 = tri + cx = (pos[v1][0] + pos[v2][0] + pos[v3][0]) / 3 + cy = (pos[v1][1] + pos[v2][1] + pos[v3][1]) / 3 + centroids.append((cx, cy)) + + # === Compute leaves and which annular face each is incident to === + # For each B_out edge: find which annular triangle contains it. + # For each B_in edge: similarly. + from collections import defaultdict + bout_edges = [frozenset({outer[i], outer[(i+1) % m]}) for i in range(m)] + bin_edges = [frozenset({inner[j], inner[(j+1) % k]}) for j in range(k)] + + # Map: edge -> list of triangle indices containing it + edge_to_tris = defaultdict(list) + for t_idx, tri in enumerate(triangles): + e1 = frozenset({tri[0], tri[1]}) + e2 = frozenset({tri[1], tri[2]}) + e3 = frozenset({tri[0], tri[2]}) + for e in (e1, e2, e3): + edge_to_tris[e].append(t_idx) + + bout_leaves = [] # list of (edge, tri_idx, leaf_position) + for e in bout_edges: + tris_with_e = edge_to_tris[e] + assert len(tris_with_e) == 1, \ + f"B_out edge {e} should be in exactly 1 annular triangle" + t_idx = tris_with_e[0] + u, v = list(e) + midx = (pos[u][0] + pos[v][0]) / 2 + midy = (pos[u][1] + pos[v][1]) / 2 + # Push outward + d = math.sqrt(midx**2 + midy**2) + push = 1.18 + leaf_pos = (midx * push / d, midy * push / d) + bout_leaves.append((e, t_idx, leaf_pos)) + + bin_leaves = [] + for e in bin_edges: + tris_with_e = edge_to_tris[e] + assert len(tris_with_e) == 1, \ + f"B_in edge {e} should be in exactly 1 annular triangle" + t_idx = tris_with_e[0] + u, v = list(e) + midx = (pos[u][0] + pos[v][0]) / 2 + midy = (pos[u][1] + pos[v][1]) / 2 + # Place the leaf aligned with the midpoint of the red edge it crosses, + # on the central-hole side, at a fixed small distance past the + # midpoint along the (d_f -> midpoint) direction. This keeps the + # leaf inside the central hole (not on the border) while making the + # leaf edge cross the red edge exactly at its midpoint. + cx, cy = centroids[t_idx] + vx, vy = midx - cx, midy - cy + norm = math.sqrt(vx * vx + vy * vy) + if norm < 1e-9: + nx, ny = 0.0, -1.0 + else: + nx, ny = vx / norm, vy / norm + offset = 0.13 # absolute distance past the midpoint + leaf_pos = (midx + nx * offset, midy + ny * offset) + bin_leaves.append((e, t_idx, leaf_pos)) + + # === Determine interior dual cycle order === + # Two annular triangles are adjacent in the dual cycle iff they share an + # interior annular edge (= an edge with both endpoints in V(B_out) ∪ V(B_in) + # whose two incident faces are both annular triangles, i.e., shared by 2 + # of our triangles). + # In a spoke-only tire each interior edge is shared by exactly 2 triangles. + tri_adj = defaultdict(set) + for e, ts in edge_to_tris.items(): + if len(ts) == 2: + t1, t2 = ts + tri_adj[t1].add(t2) + tri_adj[t2].add(t1) + + # === Draw partial tire dual === + DUAL_COLOR = '#7d3c98' # purple + LEAF_COLOR = '#e67e22' # orange + + # Interior dual edges + for t1 in range(n_tri): + for t2 in tri_adj[t1]: + if t2 <= t1: + continue + x1, y1 = centroids[t1] + x2, y2 = centroids[t2] + ax.plot([x1, x2], [y1, y2], color=DUAL_COLOR, linewidth=2.0, + linestyle='-', zorder=4) + + # Leaf edges + for e, t_idx, lpos in bout_leaves + bin_leaves: + cx, cy = centroids[t_idx] + lx, ly = lpos + ax.plot([cx, lx], [lx and cy or cy, ly], visible=False) + # use a proper plot: + ax.plot([cx, lx], [cy, ly], color=LEAF_COLOR, linewidth=1.8, + linestyle='--', zorder=4) + + # Interior dual vertices + for t_idx, (cx, cy) in enumerate(centroids): + ax.plot(cx, cy, 's', color=DUAL_COLOR, markersize=14, zorder=5) + ax.annotate(f"$d_{{{t_idx}}}$", (cx, cy), color='white', + ha='center', va='center', fontsize=7, fontweight='bold', + zorder=6) + + # Leaf vertices + for label_prefix, leaves in (('out', bout_leaves), ('in', bin_leaves)): + for i, (e, t_idx, lpos) in enumerate(leaves): + ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, + markersize=11, zorder=5) + ax.annotate(f"$\\ell^{{{label_prefix}}}_{{{i}}}$", + (lpos[0], lpos[1]), color='white', ha='center', + va='center', fontsize=6, fontweight='bold', zorder=6) + + # Legend + legend_items = [ + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8', + markersize=12, label='$B_{\\mathrm{out}}$ vertex (tire)'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#e8a8a8', + markersize=11, label='$B_{\\mathrm{in}}$ vertex (tire)'), + plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR, + markersize=12, label='$d_f$ (interior dual vertex)'), + plt.Line2D([], [], marker='D', color='w', markerfacecolor=LEAF_COLOR, + markersize=10, label='leaf vertex'), + plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0, + label='interior dual edge'), + plt.Line2D([], [], color=LEAF_COLOR, linewidth=1.8, linestyle='--', + label='leaf edge'), + ] + ax.legend(handles=legend_items, loc='upper left', + bbox_to_anchor=(1.0, 1.0), fontsize=9, frameon=False) + + ax.set_xlim(-1.35, 1.35) + ax.set_ylim(-1.35, 1.35) + ax.set_aspect('equal') + ax.axis('off') + ax.set_title( + f'Partial tire dual $D(T)$ (m={m}, k={k}, spoke-only)\n' + f'underlying tire faint; $D(T) \\cong C_{{{m+k}}} \\circ K_1$', + fontsize=12) + + plt.savefig(filename, dpi=160, bbox_inches='tight') + plt.close() + print(f"wrote {filename}") + + +def main(): + # m=6 outer, k=4 inner, no chords (spoke-only), reproducible seed + tire = random_tire(m=6, k=4, n_chords=0, seed=3) + out = os.path.join(HERE, 'partial_tire_dual_example.png') + draw_partial_tire_dual(tire, out) + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/experiments/partial_tire_dual_example.png b/papers/coloring_nested_tire_graphs/experiments/partial_tire_dual_example.png new file mode 100644 index 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The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.8\hbox {}.}}{4}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-example}{{3}{4}} \newlabel{prop:partial-tire-dual-structure}{{1.8}{4}} -\newlabel{prop:no-level-d-pinch}{{1.9}{4}} -\citation{bauerfeld-pds} +\newlabel{prop:no-level-d-pinch}{{1.9}{5}} \newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} -\newlabel{rem:tire-component-degenerate}{{1.11}{6}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{6}} +\citation{bauerfeld-pds} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\newlabel{rem:tire-component-degenerate}{{1.11}{7}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } \gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 93cecae..d4d81eb 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 17:59 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 18:37 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -207,36 +207,42 @@ File: fig_tire_example.png Graphic file (type png) Package pdftex.def Info: fig_tire_example.png used on input line 160. 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+\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png} +\caption{The partial tire dual $D(T)$ (purple squares + orange +diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ +and $k = 4$. The ten interior vertices $d_f$ at the centroids of the +annular triangles form a single $10$-cycle (solid purple); each +boundary edge of the annular region (either of $B_{\mathrm{out}}$ or +of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond) +attached to the unique annular face incident to it (dashed orange), +giving the structure $C_{10} \circ K_1$ of +Proposition~\ref{prop:partial-tire-dual-structure}.} +\label{fig:partial-tire-dual-example} +\end{figure} + \begin{proposition}[Structure of $D(T)$ when the annular triangulation is spoke-only] \label{prop:partial-tire-dual-structure}

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