coloring_nested_tire_graphs: add bridge-example partial tire dual figure beneath Fig 3

Adds fig_partial_tire_dual_bridge.png beneath the existing partial-
tire-dual figure (Figure 3).  The new figure shows a tire graph
whose inner outerplanar O has a bridge:
  B_out = triangle on {0, 1, 2};
  O     = triangle {3, 4, 5} plus pendant edge 5-6 (the bridge);
  annular triangulation with 8 triangles (constructed by hand).

Key contrast with the previous figure: because both faces incident
to the bridge are annular triangles, the bridge contributes an
INTERIOR DUAL EDGE rather than two leaves.  Consequently the
interior dual subgraph is no longer a single (n+m)-cycle (as in
Prop 1.8 for spoke-only tires) but a theta graph: two trivalent
d_f vertices connected by three internally vertex-disjoint paths.
Leaves come only from B_out (3 of them) and the three non-bridge
triangle edges of O (the inner-triangle face boundary).

Adds experiments/draw_partial_tire_dual_bridge.py to generate the
figure.

Paper grows from 8 to 9 pages.

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

papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py

@@ -0,0 +1,264 @@
+"""Draw a partial tire dual D(T) for a tire whose inner outerplanar
+graph O has a bridge.
+
+Tire construction:
+  - Outer cycle B_out: triangle on {0, 1, 2}.
+  - Inner outerplanar O on {3, 4, 5, 6}: triangle 3-4-5 plus pendant
+    edge 5-6 (the bridge of O).
+  - Annular triangulation with 8 triangles (computed by hand below).
+
+The bridge 5-6 has both incident faces in the annular region, so in
+the partial tire dual it contributes an interior dual edge (not a
+leaf).  This makes the interior dual subgraph a theta graph rather
+than a single cycle: two trivalent vertices (the two annular faces
+incident to the bridge) connected by three paths.
+"""
+import math
+import os
+import sys
+from collections import defaultdict
+
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+
+
+# Explicit vertex positions
+POS = {
+    0: (0.0, 1.0),
+    1: (-0.866, -0.5),
+    2: (0.866, -0.5),
+    3: (0.0, 0.32),
+    4: (-0.27, -0.17),
+    5: (0.13, -0.17),
+    6: (0.52, -0.07),
+}
+
+# Annular triangles (in lattice-path-like order, but here just listed)
+TRIANGLES = [
+    (0, 1, 4),     # T1 -- O-move
+    (0, 4, 3),     # T2 -- has edge 4-3 boundary (inner triangle)
+    (1, 2, 5),     # T3 -- O-move
+    (1, 5, 4),     # T4 -- inner-triangle-edge boundary
+    (2, 0, 6),     # T5 -- O-move
+    (0, 3, 6),     # T6 -- ALL INTERIOR (bridge neighbour)
+    (3, 5, 6),     # T7 -- inner-triangle-edge boundary, contains bridge
+    (5, 2, 6),     # T8 -- ALL INTERIOR (bridge neighbour)
+]
+
+# Tire edges (all)
+T_EDGES = [
+    # outer cycle (B_out)
+    (0, 1), (1, 2), (2, 0),
+    # O: triangle 3-4-5
+    (3, 4), (4, 5), (3, 5),
+    # O: pendant / bridge
+    (5, 6),
+    # spokes
+    (0, 3), (1, 4), (2, 5),
+    # diagonals from annular triangulation
+    (0, 4), (1, 5), (0, 6), (2, 6), (3, 6),
+]
+
+# Boundary edges
+B_OUT_EDGES = [(0, 1), (1, 2), (2, 0)]
+# Edges of O on inner-triangle-face boundary (i.e., NOT the bridge)
+B_IN_EDGES = [(3, 4), (4, 5), (3, 5)]
+# Bridge edge (interior to annulus)
+BRIDGE = (5, 6)
+
+
+def main():
+    # Map: edge -> list of triangle indices containing it
+    edge_to_tris = defaultdict(list)
+    for i, t in enumerate(TRIANGLES):
+        for e in (frozenset({t[0], t[1]}),
+                  frozenset({t[1], t[2]}),
+                  frozenset({t[0], t[2]})):
+            edge_to_tris[e].append(i)
+
+    # Verify each B_out edge is in exactly 1 annular triangle
+    for u, v in B_OUT_EDGES:
+        ts = edge_to_tris[frozenset({u, v})]
+        assert len(ts) == 1, f"B_out edge {u}-{v} in {ts}"
+
+    # Each B_in (= non-bridge O edge) is in exactly 1 annular triangle
+    for u, v in B_IN_EDGES:
+        ts = edge_to_tris[frozenset({u, v})]
+        assert len(ts) == 1, f"B_in edge {u}-{v} in {ts}"
+
+    # Bridge is in exactly 2 annular triangles (interior to annulus)
+    bts = edge_to_tris[frozenset(BRIDGE)]
+    print(f"Bridge {BRIDGE} is in annular triangles {bts}")
+    assert len(bts) == 2
+
+    # Interior dual subgraph: connect annular triangles sharing an edge
+    n_tri = len(TRIANGLES)
+    tri_adj = defaultdict(set)
+    for e, ts in edge_to_tris.items():
+        if len(ts) == 2:
+            tri_adj[ts[0]].add(ts[1])
+            tri_adj[ts[1]].add(ts[0])
+
+    print("Interior dual subgraph degree sequence:")
+    for i in range(n_tri):
+        print(f"  T{i+1} ({TRIANGLES[i]}): degree {len(tri_adj[i])}")
+
+    # Centroids of triangles for d_f positions
+    centroids = []
+    for t in TRIANGLES:
+        cx = (POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3
+        cy = (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3
+        centroids.append((cx, cy))
+
+    # === Plot ===
+    fig, ax = plt.subplots(figsize=(10, 9))
+
+    # Draw underlying tire edges faintly
+    outer_set = {0, 1, 2}
+    inner_set = {3, 4, 5, 6}
+    for u, v in T_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:
+            # Distinguish bridge (pendant) from triangle edges
+            if frozenset({u, v}) == frozenset(BRIDGE):
+                color = '#cc6677'; lw = 2.5  # darker red for bridge
+            else:
+                color = '#e8a8a8'; lw = 2.0
+        else:
+            color = '#cccccc'; lw = 1.0
+        ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
+
+    # Tire vertices
+    for v in [0, 1, 2]:
+        x, y = POS[v]
+        ax.plot(x, y, 'o', color='#a8c9e8', markersize=15, zorder=2)
+        ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
+                    fontsize=9, fontweight='bold', zorder=3)
+    for v in [3, 4, 5, 6]:
+        x, y = POS[v]
+        ax.plot(x, y, 'o', color='#e8a8a8', markersize=13, zorder=2)
+        ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
+                    fontsize=8, fontweight='bold', zorder=3)
+
+    DUAL_COLOR = '#7d3c98'
+    LEAF_COLOR = '#e67e22'
+    BRIDGE_DUAL_COLOR = '#cc4444'   # color the bridge's dual edge differently
+
+    # Interior dual edges
+    for i in range(n_tri):
+        for j in tri_adj[i]:
+            if j <= i:
+                continue
+            x1, y1 = centroids[i]; x2, y2 = centroids[j]
+            # Find shared edge to colour-code bridge edge
+            ti_es = {frozenset({TRIANGLES[i][a], TRIANGLES[i][b]})
+                     for a, b in [(0, 1), (1, 2), (0, 2)]}
+            tj_es = {frozenset({TRIANGLES[j][a], TRIANGLES[j][b]})
+                     for a, b in [(0, 1), (1, 2), (0, 2)]}
+            shared = (ti_es & tj_es).pop()
+            if shared == frozenset(BRIDGE):
+                ec, lw = BRIDGE_DUAL_COLOR, 3.0
+            else:
+                ec, lw = DUAL_COLOR, 2.0
+            ax.plot([x1, x2], [y1, y2], color=ec, linewidth=lw, zorder=4)
+
+    # Leaves for B_out edges
+    for u, v in B_OUT_EDGES:
+        ti = edge_to_tris[frozenset({u, v})][0]
+        midx = (POS[u][0] + POS[v][0]) / 2
+        midy = (POS[u][1] + POS[v][1]) / 2
+        d = math.sqrt(midx**2 + midy**2)
+        push = 1.20
+        lpos = (midx * push / d, midy * push / d)
+        cx, cy = centroids[ti]
+        ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
+                linewidth=1.5, linestyle='--', zorder=4)
+        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
+                zorder=5)
+        ax.annotate(f"$\\ell^{{\\mathrm{{out}}}}_{{{u},{v}}}$", lpos,
+                    color='white', ha='center', va='center', fontsize=6,
+                    fontweight='bold', zorder=6)
+
+    # Leaves for non-bridge B_in edges (= the 3 inner triangle edges)
+    for u, v in B_IN_EDGES:
+        ti = edge_to_tris[frozenset({u, v})][0]
+        midx = (POS[u][0] + POS[v][0]) / 2
+        midy = (POS[u][1] + POS[v][1]) / 2
+        cx, cy = centroids[ti]
+        vx, vy = midx - cx, midy - cy
+        norm = math.sqrt(vx*vx + vy*vy)
+        offset = 0.10
+        if norm > 1e-9:
+            nx, ny = vx / norm, vy / norm
+        else:
+            nx, ny = 0, -1
+        lpos = (midx + nx * offset, midy + ny * offset)
+        ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR,
+                linewidth=1.5, linestyle='--', zorder=4)
+        ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11,
+                zorder=5)
+        ax.annotate(f"$\\ell^{{\\mathrm{{in}}}}_{{{u},{v}}}$", lpos,
+                    color='white', ha='center', va='center', fontsize=6,
+                    fontweight='bold', zorder=6)
+
+    # Interior dual vertices d_f, highlighting the two degree-3
+    # vertices in a darker color (these are the trivalent vertices
+    # of the theta graph that the interior dual forms).
+    for ti, (cx, cy) in enumerate(centroids):
+        is_deg3 = (len(tri_adj[ti]) == 3)
+        marker_color = '#5a2273' if is_deg3 else DUAL_COLOR
+        ax.plot(cx, cy, 's', color=marker_color, markersize=15, zorder=5)
+        ax.annotate(f"$d_{{{ti}}}$", (cx, cy), color='white', ha='center',
+                    va='center', fontsize=7, fontweight='bold', zorder=6)
+
+    # Legend
+    legend_items = [
+        plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8',
+                   markersize=12, label='$B_{\\mathrm{out}}$ vertex'),
+        plt.Line2D([], [], marker='o', color='w', markerfacecolor='#e8a8a8',
+                   markersize=11, label='$V(O)$ vertex'),
+        plt.Line2D([], [], color='#a8c9e8', linewidth=2.0,
+                   label='$B_{\\mathrm{out}}$ edge'),
+        plt.Line2D([], [], color='#e8a8a8', linewidth=2.0,
+                   label='triangle edge of $O$'),
+        plt.Line2D([], [], color='#cc6677', linewidth=2.5,
+                   label='bridge of $O$ (= pendant)'),
+        plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR,
+                   markersize=12, label='$d_f$ (degree 2 in interior dual)'),
+        plt.Line2D([], [], marker='s', color='w', markerfacecolor='#5a2273',
+                   markersize=12,
+                   label='$d_f$ (degree 3 -- theta-graph trivalent)'),
+        plt.Line2D([], [], marker='D', color='w', markerfacecolor=LEAF_COLOR,
+                   markersize=10, label='leaf'),
+        plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0,
+                   label='interior dual edge'),
+        plt.Line2D([], [], color=BRIDGE_DUAL_COLOR, linewidth=3.0,
+                   label='dual of the bridge'),
+        plt.Line2D([], [], color=LEAF_COLOR, linewidth=1.5, linestyle='--',
+                   label='leaf edge'),
+    ]
+    ax.legend(handles=legend_items, loc='upper left',
+              bbox_to_anchor=(1.02, 1.0), fontsize=8, frameon=False)
+
+    ax.set_xlim(-1.30, 1.50)
+    ax.set_ylim(-1.20, 1.20)
+    ax.set_aspect('equal')
+    ax.axis('off')
+    ax.set_title(
+        'Partial tire dual $D(T)$ when $O$ has a bridge\n'
+        '$O$ = triangle $\\{3,4,5\\}$ + pendant edge $5$--$6$;\n'
+        'interior dual is a theta graph (2 deg-3 vertices + 3 paths), '
+        'not a cycle.',
+        fontsize=11)
+
+    HERE = os.path.dirname(os.path.abspath(__file__))
+    out = os.path.join(HERE, 'partial_tire_dual_bridge.png')
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/paper.aux

@@ -11,19 +11,21 @@
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces 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\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}}
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge. Here $B_{\mathrm  {out}}$ is a triangle on $\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge $5$--$6$ (the bridge of $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph  {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph (two trivalent vertices $d_5, d_7$ connected by three internally vertex-disjoint paths in $D(T)$). Leaves come only from $B_{\mathrm  {out}}$ ($n = 3$ leaves) and from the three non-bridge edges of $O$ (the three triangle edges of the inner triangle).}}{5}{}\protected@file@percent }
+\newlabel{fig:partial-tire-dual-bridge}{{4}{5}}
 \newlabel{prop:no-level-d-pinch}{{1.9}{5}}
-\newlabel{lem:tire-component}{{1.10}{5}}
 \citation{bauerfeld-pds}
+\newlabel{lem:tire-component}{{1.10}{6}}
 \citation{bauerfeld-pds}
 \newlabel{rem:tire-component-degenerate}{{1.11}{7}}
-\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
+\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}}
-\newlabel{prop:edge-vertex-bijection}{{1.13}{7}}
+\newlabel{prop:edge-vertex-bijection}{{1.13}{8}}
+\newlabel{rem:edge-vertex-corollary}{{1.14}{8}}
 \bibcite{bauerfeld-pds}{1}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\newlabel{rem:edge-vertex-corollary}{{1.14}{8}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
+\gdef \@abspage@last{9}
-\gdef \@abspage@last{8}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -235,6 +235,25 @@ Proposition~\ref{prop:partial-tire-dual-structure}.}
 \label{fig:partial-tire-dual-example}
 \end{figure}
 
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual_bridge.png}
+\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
+$O$ has a bridge.  Here $B_{\mathrm{out}}$ is a triangle on
+$\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge
+$5$--$6$ (the bridge of $O$).  Because both faces incident to the
+bridge are annular triangles, the bridge contributes an
+\emph{interior dual edge} (highlighted in red) rather than two
+leaves; consequently the interior dual subgraph is no longer the
+single $(n+m)$-cycle of
+Proposition~\ref{prop:partial-tire-dual-structure}, but a theta
+graph (two trivalent vertices $d_5, d_7$ connected by three
+internally vertex-disjoint paths in $D(T)$).  Leaves come only from
+$B_{\mathrm{out}}$ ($n = 3$ leaves) and from the three non-bridge
+edges of $O$ (the three triangle edges of the inner triangle).}
+\label{fig:partial-tire-dual-bridge}
+\end{figure}
+
 \begin{proposition}[Structure of $D(T)$ when the annular triangulation
 is spoke-only]
 \label{prop:partial-tire-dual-structure}