coloring_nested_tire_graphs: add figure of the complete tire dual D*(T)

Adds fig_complete_tire_dual.png next to Definition 1.13 of the
complete tire dual.  The figure uses the same m=6, k=4 spoke-only
tire as the partial-tire-dual figure (so the two figures can be
directly compared), with the n=6 outer leaves merged into a single
outer-face vertex v_out (blue hexagon, drawn outside the tire) of
degree 6, and the m=4 inner leaves merged into a single inner-face
vertex v_in (red hexagon, drawn at the centre of the inner cycle)
of degree 4.

Generator: experiments/draw_complete_tire_dual.py.

Paper grows from 8 to 9 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/coloring_nested_tire_graphs/experiments/draw_complete_tire_dual.py

@@ -0,0 +1,199 @@
+"""Draw the complete tire dual D*(T) on top of a small tire graph.
+
+For a spoke-only tire with 2-connected O = C_k (no chords), D*(T) is
+obtained from D(T) by merging:
+  - all B_out-leaves into a single outer-face vertex v_out (degree n);
+  - the B_in-leaves into a single inner-face vertex v_in (degree k),
+    because O has exactly one bounded interior face.
+
+Equivalently, D*(T) is the planar dual of T.
+"""
+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_complete_tire_dual(tire, filename):
+    m, k = tire['m'], tire['k']
+    outer, inner = tire['outer'], tire['inner']
+    triangles = tire['triangles']
+
+    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)
+
+    n_tri = len(triangles)
+    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))
+
+    # Identify which triangles are incident to a B_out edge (O-move type)
+    # vs incident to a B_in edge (I-move type).
+    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)}
+
+    bout_incident = []   # triangles incident to a B_out edge
+    bin_incident = []    # triangles incident to a B_in edge
+    for t_idx, tri in enumerate(triangles):
+        tri_edges = {frozenset({tri[0], tri[1]}),
+                     frozenset({tri[1], tri[2]}),
+                     frozenset({tri[0], tri[2]})}
+        if tri_edges & bout_edges:
+            bout_incident.append(t_idx)
+        if tri_edges & bin_edges:
+            bin_incident.append(t_idx)
+
+    print(f"O-move triangles (B_out incident): {bout_incident}")
+    print(f"I-move triangles (B_in incident): {bin_incident}")
+
+    # Place v_out outside the outer cycle (far to the upper right)
+    # and v_in at the origin (centroid of inner cycle).
+    v_out_pos = (1.55, 0.0)   # well outside the outer cycle
+    v_in_pos = (0.0, 0.0)     # center of inner cycle
+
+    # Identify dual cycle (interior dual edges)
+    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)
+
+    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 complete tire dual ===
+    DUAL_COLOR = '#7d3c98'   # purple
+    OUT_COLOR = '#1f77b4'    # blue (for v_out edges)
+    IN_COLOR = '#d62728'     # red (for v_in edges)
+
+    # Interior dual edges (d_f to d_f)
+    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,
+                    zorder=4)
+
+    # v_out edges (to all O-move d_f's)
+    for t_idx in bout_incident:
+        cx, cy = centroids[t_idx]
+        ax.plot([cx, v_out_pos[0]], [cy, v_out_pos[1]],
+                color=OUT_COLOR, linewidth=1.6, linestyle='--', zorder=4)
+
+    # v_in edges (to all I-move d_f's)
+    for t_idx in bin_incident:
+        cx, cy = centroids[t_idx]
+        ax.plot([cx, v_in_pos[0]], [cy, v_in_pos[1]],
+                color=IN_COLOR, linewidth=1.6, linestyle='--', zorder=4)
+
+    # Interior dual vertices d_f
+    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)
+
+    # v_out (outer-face vertex)
+    ax.plot(v_out_pos[0], v_out_pos[1], 'h', color=OUT_COLOR,
+            markersize=20, zorder=5)
+    ax.annotate("$v_{\\mathrm{out}}$", v_out_pos, color='white',
+                ha='center', va='center', fontsize=9, fontweight='bold',
+                zorder=6)
+
+    # v_in (inner-face vertex)
+    ax.plot(v_in_pos[0], v_in_pos[1], 'h', color=IN_COLOR,
+            markersize=18, zorder=5)
+    ax.annotate("$v_{\\mathrm{in}}$", v_in_pos, color='white',
+                ha='center', va='center', fontsize=8, 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='$d_f$ (annular face dual vertex)'),
+        plt.Line2D([], [], marker='h', color='w', markerfacecolor=OUT_COLOR,
+                   markersize=14,
+                   label='$v_{\\mathrm{out}}$ (outer-face vertex, deg ${%d}$)' % m),
+        plt.Line2D([], [], marker='h', color='w', markerfacecolor=IN_COLOR,
+                   markersize=13,
+                   label='$v_{\\mathrm{in}}$ (inner-face vertex, deg ${%d}$)' % k),
+        plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0,
+                   label='interior dual edge ($d_f$–$d_{f^\\prime}$)'),
+        plt.Line2D([], [], color=OUT_COLOR, linewidth=1.6, linestyle='--',
+                   label='dual edge to $v_{\\mathrm{out}}$'),
+        plt.Line2D([], [], color=IN_COLOR, linewidth=1.6, linestyle='--',
+                   label='dual edge to $v_{\\mathrm{in}}$'),
+    ]
+    ax.legend(handles=legend_items, loc='upper left',
+              bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False)
+
+    ax.set_xlim(-1.35, 2.05)
+    ax.set_ylim(-1.35, 1.35)
+    ax.set_aspect('equal')
+    ax.axis('off')
+    ax.set_title(
+        f'Complete tire dual $D^{{*}}(T)$ (m={m}, k={k}, spoke-only)\n'
+        f'underlying tire faint; $|V| = {n_tri + 2}$, $|E| = {2*n_tri}$',
+        fontsize=12)
+
+    plt.savefig(filename, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {filename}")
+
+
+def main():
+    tire = random_tire(m=6, k=4, n_chords=0, seed=3)
+    out = os.path.join(HERE, 'complete_tire_dual_example.png')
+    draw_complete_tire_dual(tire, out)
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/paper.aux

@@ -19,6 +19,11 @@
 \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
 \newlabel{def:complete-tire-dual}{{1.13}{7}}
 \citation{Tait1880}
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The complete tire dual $D^{\ast }(T)$ (purple squares + face hexagons) drawn on top of the same $m = 6$, $k = 4$ spoke-only tire graph as Figure\nonbreakingspace 3\hbox {}. The ten annular-face vertices $d_f$ form a $10$-cycle (solid purple); the $n$ outer leaves of $D(T)$ have been merged into a single outer-face vertex $v_{\mathrm  {out}}$ (blue hexagon, drawn outside the tire) of degree $n = 6$, and the $m$ inner leaves into a single inner-face vertex $v_{\mathrm  {in}}$ (red hexagon, at the centre of the inner cycle) of degree $m = 4$. Total $|V(D^{\ast }(T))| = 12$ and $|E(D^{\ast }(T))| = 20$.}}{8}{}\protected@file@percent }
+\newlabel{fig:complete-tire-dual-example}{{4}{8}}
+\newlabel{prop:tait-tire-complete}{{1.14}{8}}
+\newlabel{rem:tait-construction}{{1.15}{8}}
+\newlabel{rem:tait-octahedron}{{1.16}{8}}
 \bibcite{Tait1880}{1}
 \bibcite{bauerfeld-pds}{2}
 \newlabel{tocindent-1}{0pt}
@@ -26,8 +31,5 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\newlabel{prop:tait-tire-complete}{{1.14}{8}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
-\newlabel{rem:tait-construction}{{1.15}{8}}
+\gdef \@abspage@last{9}
-\newlabel{rem:tait-octahedron}{{1.16}{8}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
-\gdef \@abspage@last{8}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -508,6 +508,22 @@ vertex per face of $T$ (annular triangle, outer face, or bounded
 interior face of $O$), one dual edge per edge of $T$.
 \end{definition}
 
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.82\textwidth]{fig_complete_tire_dual.png}
+\caption{The complete tire dual $D^{\ast}(T)$ (purple squares + face
+hexagons) drawn on top of the same $m = 6$, $k = 4$ spoke-only tire
+graph as Figure~\ref{fig:partial-tire-dual-example}.  The ten
+annular-face vertices $d_f$ form a $10$-cycle (solid purple); the
+$n$ outer leaves of $D(T)$ have been merged into a single outer-face
+vertex $v_{\mathrm{out}}$ (blue hexagon, drawn outside the tire) of
+degree $n = 6$, and the $m$ inner leaves into a single inner-face
+vertex $v_{\mathrm{in}}$ (red hexagon, at the centre of the inner
+cycle) of degree $m = 4$.  Total $|V(D^{\ast}(T))| = 12$ and
+$|E(D^{\ast}(T))| = 20$.}
+\label{fig:complete-tire-dual-example}
+\end{figure}
+
 \begin{proposition}[Tait correspondence on the complete tire dual]
 \label{prop:tait-tire-complete}
 Let $T$ be a tire graph.  Then the number of non-equivalent proper