coloring_nested_tire_graphs: add fiber decomposition note for T'_{f'} edge colorings

Decomposes P_e(T'_{f'}, k) into fibers over spoke configurations σ,
specializes to the menagerie §6 formula when there are no spokes, and
sets up the cross-tire compatibility step on shared boundary spokes.
Includes brute-force-verified worked example on C_4 with four pendants.

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

papers/coloring_nested_tire_graphs/notes/draw_fiber_c4_example.py

@@ -0,0 +1,162 @@
+"""Worked example for fiber_decomposition.tex.
+Three panels: C_4 with 4 pendant spokes, showing
+  (a) sigma = (1,1,1,1) -- fiber count 2 (two alternating cycle colorings).
+  (b) sigma = (1,2,1,2) -- infeasible, fiber count 0.
+  (c) sigma = (1,1,2,2) -- fiber count 1 (unique extension).
+"""
+import math
+import os
+import matplotlib.pyplot as plt
+from itertools import product
+
+
+COLORS = {
+    1: '#d62728',  # red
+    2: '#1f77b4',  # blue
+    3: '#2ca02c',  # green
+}
+COLOR_NAMES = {1: 'red', 2: 'blue', 3: 'green'}
+
+
+def square_pos(scale=1.0):
+    # v_0 right, v_1 top, v_2 left, v_3 bottom -- a square
+    return {
+        0: (scale, 0),
+        1: (0, scale),
+        2: (-scale, 0),
+        3: (0, -scale),
+    }
+
+
+def ext_pos(scale=1.0, radial=1.9):
+    pos = square_pos(scale)
+    out = {}
+    for v, (x, y) in pos.items():
+        nx, ny = x / math.hypot(x, y), y / math.hypot(x, y)
+        out[v] = (radial * nx, radial * ny)
+    return out
+
+
+def enumerate_fiber(sigma):
+    """Brute-force: enumerate all cycle-edge colorings (c_e0, c_e1, c_e2, c_e3)
+    in [3]^4 satisfying:
+      proper edge coloring at each v_i (its three incident edges all distinct)
+      and the pendant at v_i is sigma[i].
+    e_i goes from v_i to v_{i+1}.  v_i is incident to e_{i-1} (mod 4) and e_i.
+    Returns the list of valid (c_e0, c_e1, c_e2, c_e3) tuples.
+    """
+    valid = []
+    for ce in product([1, 2, 3], repeat=4):
+        ok = True
+        for i in range(4):
+            ep = ce[(i - 1) % 4]
+            en = ce[i]
+            s = sigma[i]
+            if len({ep, en, s}) != 3:
+                ok = False
+                break
+        if ok:
+            valid.append(ce)
+    return valid
+
+
+def draw_panel(ax, sigma, title):
+    pos = square_pos()
+    ext = ext_pos()
+    # Cycle edges (drawn light first, overdrawn in colored if there is a unique
+    # extension; if there are multiple extensions show grey-with-letters)
+    fiber = enumerate_fiber(sigma)
+
+    # Cycle edges: e_i from v_i to v_{i+1}
+    for i in range(4):
+        u, v = i, (i + 1) % 4
+        x1, y1 = pos[u]; x2, y2 = pos[v]
+        if len(fiber) == 1:
+            ax.plot([x1, x2], [y1, y2], color=COLORS[fiber[0][i]],
+                    linewidth=4.0, zorder=1)
+        elif len(fiber) >= 2:
+            # show all extensions as parallel offset lines
+            colors_for_edge = sorted({f[i] for f in fiber})
+            n = len(colors_for_edge)
+            # tangent perpendicular to edge
+            dx, dy = x2 - x1, y2 - y1
+            ln = math.hypot(dx, dy)
+            px, py = -dy / ln, dx / ln  # unit normal
+            spacing = 0.08
+            offsets = [(j - (n - 1) / 2) * spacing for j in range(n)]
+            for off, c in zip(offsets, colors_for_edge):
+                ax.plot([x1 + off * px, x2 + off * px],
+                        [y1 + off * py, y2 + off * py],
+                        color=COLORS[c], linewidth=3.0, zorder=1)
+        else:
+            # infeasible: dashed grey
+            ax.plot([x1, x2], [y1, y2], color='#bbbbbb', linewidth=2.5,
+                    linestyle=(0, (4, 3)), zorder=1)
+
+    # Spoke edges: solid in sigma's color
+    for i in range(4):
+        x1, y1 = pos[i]; x2, y2 = ext[i]
+        ax.plot([x1, x2], [y1, y2], color=COLORS[sigma[i]], linewidth=4.0,
+                zorder=1)
+
+    # Cycle vertices
+    for v, (x, y) in pos.items():
+        ax.plot(x, y, 'o', color='#1f3f70', markersize=18, zorder=2)
+        ax.annotate(f"$v_{v}$", (x, y), color='white', ha='center',
+                    va='center', fontsize=10, fontweight='bold', zorder=3)
+    # Spoke vertices
+    for v, (x, y) in ext.items():
+        ax.plot(x, y, 's', color='#666666', markersize=15, zorder=2)
+        ax.annotate(f"$u_{v}$", (x, y), color='white', ha='center',
+                    va='center', fontsize=8, fontweight='bold', zorder=3)
+
+    # Subtitle with fiber count
+    n = len(fiber)
+    status = (f"$N(\\sigma) = {n}$ "
+              + ("(infeasible)" if n == 0
+                 else "(unique extension)" if n == 1
+                 else "(multiple extensions, shown as parallel strands)"))
+    ax.set_xlim(-2.5, 2.5)
+    ax.set_ylim(-2.5, 2.5)
+    ax.set_aspect('equal')
+    ax.axis('off')
+    ax.set_title(title + "\n" + status, fontsize=10)
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    out = os.path.join(here, 'fig_fiber_c4_example.png')
+    fig, axes = plt.subplots(1, 3, figsize=(13.5, 5.0))
+
+    draw_panel(axes[0], (1, 1, 1, 1),
+               r"$\sigma = (1,1,1,1)$ (all spokes red)")
+    draw_panel(axes[1], (1, 2, 1, 2),
+               r"$\sigma = (1,2,1,2)$ (alternating)")
+    draw_panel(axes[2], (1, 1, 2, 2),
+               r"$\sigma = (1,1,2,2)$")
+
+    # legend
+    legend_items = [
+        plt.Line2D([], [], color=COLORS[1], lw=3,
+                   label='color 1 (red)'),
+        plt.Line2D([], [], color=COLORS[2], lw=3,
+                   label='color 2 (blue)'),
+        plt.Line2D([], [], color=COLORS[3], lw=3,
+                   label='color 3 (green)'),
+    ]
+    fig.legend(handles=legend_items, loc='lower center', ncol=3,
+               frameon=False, fontsize=10, bbox_to_anchor=(0.5, -0.02))
+    plt.tight_layout()
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+    # Also print the actual enumeration so we know our claims in the .tex
+    # match the brute-force counts.
+    for sigma in [(1, 1, 1, 1), (1, 2, 1, 2), (1, 1, 2, 2)]:
+        f = enumerate_fiber(sigma)
+        print(f"sigma = {sigma}: {len(f)} extensions: {f}")
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/notes/draw_fiber_setup.py

@@ -0,0 +1,99 @@
+"""Two-panel figure for fiber_decomposition.tex:
+LEFT  -- the core T'_ann = theta(1,3,3) alone (what menagerie sec.6 counts).
+RIGHT -- the face connector T'_{f'} = core + 4 spoke pendants (what we want)."""
+import math
+import os
+import matplotlib.pyplot as plt
+
+
+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
+
+
+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_core(ax):
+    pos = hex_pos()
+    # cycle
+    for u, v in CYCLE_EDGES:
+        x1, y1 = pos[u]; x2, y2 = pos[v]
+        ax.plot([x1, x2], [y1, y2], color='#1f3f70', linewidth=3.0, zorder=1)
+    # chord
+    x1, y1 = pos[CHORD[0]]; x2, y2 = pos[CHORD[1]]
+    ax.plot([x1, x2], [y1, y2], color='#1f3f70', linewidth=3.0,
+            linestyle=(0, (3, 2)), zorder=1)
+    # vertices
+    for v, (x, y) in pos.items():
+        ax.plot(x, y, 'o', color='#1f3f70', markersize=16, zorder=2)
+        ax.annotate(f"$v_{v}$", (x, y), color='white', ha='center',
+                    va='center', fontsize=9, fontweight='bold', zorder=3)
+    ax.set_xlim(-2.2, 2.2)
+    ax.set_ylim(-2.2, 2.2)
+    ax.set_aspect('equal'); ax.axis('off')
+    ax.set_title(r"Core $T'_{\mathrm{ann}} = \theta(1,3,3)$" + "\n" +
+                 r"(menagerie \S6 counts this)", fontsize=10)
+
+
+def draw_connector(ax):
+    pos = hex_pos()
+    ext = ext_pos()
+    # core (kept dark blue)
+    for u, v in CYCLE_EDGES:
+        x1, y1 = pos[u]; x2, y2 = pos[v]
+        ax.plot([x1, x2], [y1, y2], color='#1f3f70', linewidth=3.0, zorder=1)
+    x1, y1 = pos[CHORD[0]]; x2, y2 = pos[CHORD[1]]
+    ax.plot([x1, x2], [y1, y2], color='#1f3f70', linewidth=3.0,
+            linestyle=(0, (3, 2)), zorder=1)
+    # spoke edges (orange to stand out)
+    for v in (1, 2, 4, 5):
+        x1, y1 = pos[v]; x2, y2 = ext[v]
+        ax.plot([x1, x2], [y1, y2], color='#e08000', linewidth=3.0, zorder=1)
+    # cycle vertices
+    for v, (x, y) in pos.items():
+        ax.plot(x, y, 'o', color='#1f3f70', markersize=16, zorder=2)
+        ax.annotate(f"$v_{v}$", (x, y), color='white', ha='center',
+                    va='center', fontsize=9, fontweight='bold', zorder=3)
+    # spoke vertices
+    for v in (1, 2, 4, 5):
+        x, y = ext[v]
+        ax.plot(x, y, 's', color='#e08000', markersize=14, zorder=2)
+        ax.annotate(f"$u_{v}$", (x, y), color='white', ha='center',
+                    va='center', fontsize=8, fontweight='bold', zorder=3)
+    ax.set_xlim(-2.4, 2.4)
+    ax.set_ylim(-2.4, 2.4)
+    ax.set_aspect('equal'); ax.axis('off')
+    ax.set_title(r"Connector $T'_{f'}$ = core + 4 spoke pendants" + "\n" +
+                 r"(spoke edges in orange; this is what we actually want)",
+                 fontsize=10)
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    out = os.path.join(here, 'fig_fiber_setup.png')
+    fig, axes = plt.subplots(1, 2, figsize=(11.0, 5.0))
+    draw_core(axes[0])
+    draw_connector(axes[1])
+    plt.tight_layout()
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/notes/draw_nesting_compatibility.py

@@ -0,0 +1,155 @@
+"""Two-tire nesting picture for fiber_decomposition.tex.
+
+We draw two concentric annuli (the outer tire and inner tire) sharing
+a common cycle gamma in the middle.  Each annulus is triangulated, and
+each triangular face has a dual vertex.  The shared cycle gamma carries
+dual vertices that are annular for both tires; for one vertex v on gamma
+we highlight the single G'-edge that crosses the shared cycle -- the
+'spoke' edge that is an outer spoke of the inner tire and an inner spoke
+of the outer tire simultaneously."""
+import math
+import os
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    out = os.path.join(here, 'fig_nesting_compatibility.png')
+
+    fig, ax = plt.subplots(figsize=(9.0, 7.5))
+
+    # Three radii: outer boundary of outer tire, shared cycle gamma, inner boundary
+    R_OUT = 3.0
+    R_GAMMA = 2.0
+    R_IN = 1.0
+
+    n = 8  # number of vertices on each cycle (just for the picture)
+
+    def ring(R):
+        return [(R * math.cos(2 * math.pi * i / n + math.pi / 2),
+                 R * math.sin(2 * math.pi * i / n + math.pi / 2))
+                for i in range(n)]
+
+    out_pts = ring(R_OUT)
+    gamma_pts = ring(R_GAMMA)
+    in_pts = ring(R_IN)
+
+    # Light shaded annular regions
+    outer_annulus = patches.Wedge((0, 0), R_OUT, 0, 360, width=R_OUT - R_GAMMA,
+                                  facecolor='#e6f0ff', edgecolor='none', zorder=0)
+    inner_annulus = patches.Wedge((0, 0), R_GAMMA, 0, 360, width=R_GAMMA - R_IN,
+                                  facecolor='#fff2e0', edgecolor='none', zorder=0)
+    ax.add_patch(outer_annulus)
+    ax.add_patch(inner_annulus)
+
+    # Cycles
+    for pts, color in [(out_pts, '#333'), (gamma_pts, '#1f3f70'), (in_pts, '#333')]:
+        for i in range(n):
+            x1, y1 = pts[i]; x2, y2 = pts[(i + 1) % n]
+            ax.plot([x1, x2], [y1, y2], color=color, linewidth=2.0, zorder=1)
+
+    # Triangulate each annulus (simple zig-zag)
+    for ring_a, ring_b in [(out_pts, gamma_pts), (gamma_pts, in_pts)]:
+        for i in range(n):
+            x1, y1 = ring_a[i]; x2, y2 = ring_b[i]
+            ax.plot([x1, x2], [y1, y2], color='#999', linewidth=0.8, zorder=1)
+            x3, y3 = ring_b[(i + 1) % n]
+            ax.plot([x1, x2], [y1, y2], color='#999', linewidth=0.8, zorder=1)
+            ax.plot([ring_a[i][0], ring_b[(i + 1) % n][0]],
+                    [ring_a[i][1], ring_b[(i + 1) % n][1]],
+                    color='#999', linewidth=0.8, zorder=1)
+
+    # Cycle vertices on gamma
+    for (x, y) in gamma_pts:
+        ax.plot(x, y, 'o', color='#1f3f70', markersize=7, zorder=2)
+
+    # Pick a focal vertex v on gamma and draw the local G'-picture:
+    # the two annular faces of the outer tire incident to v give the
+    # cycle edges of T'^outer_ann at v, the two annular faces of the
+    # inner tire incident to v give the cycle edges of T'^inner_ann at v,
+    # and the single 'crossing' G'-edge at v ... well, in the spoke-only
+    # case the dual vertex at v has degree 3 in G' and one of those edges
+    # crosses gamma.
+    #
+    # For the picture we mark the dual vertex at v with a small star and
+    # draw its three G'-edges in distinct colors.
+    v_idx = 0
+    vx, vy = gamma_pts[v_idx]
+    # We draw a dual vertex slightly offset (since v on gamma is a *vertex*
+    # of G, not a dual; really we want to mark an annular face dual on each
+    # side of gamma).  Skip the dual-vertex picture and instead just
+    # highlight the shared cycle gamma and one focal vertex v with arrows
+    # pointing to its outer- and inner-tire roles.
+
+    ax.plot(vx, vy, 'o', color='#d62728', markersize=14, zorder=3)
+    ax.annotate("$v \\in V(\\gamma)$", (vx, vy), xytext=(vx + 0.8, vy + 0.3),
+                fontsize=11, color='#d62728',
+                arrowprops=dict(arrowstyle='->', color='#d62728', lw=1.2))
+
+    # The 'shared spoke' for v: in the spoke-only Remark 1.16.1 picture,
+    # T^outer_ann's cycle near v carries one G'-edge crossing inward
+    # (to a face inside gamma == an inner spoke of T^outer_{f'}), and
+    # T^inner_ann's cycle near v carries one G'-edge crossing outward
+    # (to a face outside gamma == an outer spoke of T^inner_{f'}).
+    # In the spoke-only case these are the SAME G'-edge.
+    #
+    # Draw it as a heavy orange line crossing gamma at v.
+    r_outer_face = (R_GAMMA + R_OUT) / 2  # mid-radius of outer annulus
+    r_inner_face = (R_GAMMA + R_IN) / 2   # mid-radius of inner annulus
+    theta = math.pi / 2  # vertex 0 is at the top
+    px_out = r_outer_face * math.cos(theta)
+    py_out = r_outer_face * math.sin(theta)
+    px_in = r_inner_face * math.cos(theta)
+    py_in = r_inner_face * math.sin(theta)
+    ax.plot([px_out, px_in], [py_out, py_in], color='#e08000', linewidth=4.0,
+            zorder=2)
+    ax.plot(px_out, py_out, 's', color='#e08000', markersize=14, zorder=3)
+    ax.plot(px_in, py_in, 's', color='#e08000', markersize=14, zorder=3)
+    ax.annotate("outer-tire\nannular face $f$",
+                (px_out, py_out), xytext=(2.4, 2.6), fontsize=9,
+                color='#a05000',
+                arrowprops=dict(arrowstyle='->', color='#a05000', lw=1.0))
+    ax.annotate("inner-tire\nannular face $f'$",
+                (px_in, py_in), xytext=(1.5, 0.6), fontsize=9,
+                color='#a05000',
+                arrowprops=dict(arrowstyle='->', color='#a05000', lw=1.0))
+    ax.annotate("shared $G'$-edge at $v$\n(spoke from both sides)",
+                ((px_out + px_in) / 2, (py_out + py_in) / 2),
+                xytext=(-3.6, 1.8), fontsize=10, color='#e08000',
+                arrowprops=dict(arrowstyle='->', color='#e08000', lw=1.2))
+
+    # Label the cycles
+    ax.text(0, R_OUT + 0.25, r"$B_{\mathrm{out}}$ of outer tire",
+            ha='center', fontsize=9, color='#333')
+    ax.text(0, -R_IN - 0.35, r"$B_{\mathrm{in}}$ of inner tire",
+            ha='center', fontsize=9, color='#333')
+    ax.text(-R_GAMMA - 0.45, 0, r"$\gamma$", ha='right', va='center',
+            fontsize=14, color='#1f3f70', fontweight='bold')
+
+    # Region labels
+    ax.text(0, (R_OUT + R_GAMMA) / 2 + 0.1, "outer tire",
+            ha='center', fontsize=10, color='#1f3f70',
+            bbox=dict(boxstyle='round,pad=0.3', facecolor='white',
+                      edgecolor='none', alpha=0.7))
+    ax.text(0, -(R_GAMMA + R_IN) / 2 - 0.1, "inner tire",
+            ha='center', fontsize=10, color='#a05000',
+            bbox=dict(boxstyle='round,pad=0.3', facecolor='white',
+                      edgecolor='none', alpha=0.7))
+
+    ax.set_xlim(-4.2, 4.2)
+    ax.set_ylim(-4.2, 4.2)
+    ax.set_aspect('equal')
+    ax.axis('off')
+    ax.set_title("Nested tires sharing cycle $\\gamma$:\n"
+                 "the spoke from $v \\in V(\\gamma)$ is shared between "
+                 "$T'^{\\mathrm{outer}}_{f'}$ and $T'^{\\mathrm{inner}}_{f'}$",
+                 fontsize=11)
+
+    plt.savefig(out, dpi=160, bbox_inches='tight')
+    plt.close()
+    print(f"wrote {out}")
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/notes/fiber_decomposition.aux

@@ -0,0 +1,14 @@
+\relax 
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Anatomy of a tire annular face connector $T'_{f'}$ in the spoke-only case. \textbf  {Left:} the core $T'_{\mathrm  {ann}} = C_n + M$ (here $\theta (1,3,3)$), the object the menagerie \S  6 formula counts. \textbf  {Right:} $T'_{f'}$ adds one pendant spoke edge for each non-chord boundary vertex; the spoke vertex is a dual vertex of a face of $G$ \emph  {outside} the tire annulus. In a nested-tire setting, that outside face belongs to the next tire over.\relax }}{1}{}\protected@file@percent }
+\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
+\newlabel{fig:fiber-setup}{{1}{1}}
+\@writefile{toc}{\contentsline {paragraph}{What the fiber count really is.}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Special case: $E_S = \emptyset $.}{2}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three spoke configurations $\sigma $ on $C_4$ with four pendant spokes, with cycle vertices labelled $v_0, v_1, v_2, v_3$ and spoke vertices $u_0, u_1, u_2, u_3$. \textbf  {Left:} the constant configuration $\sigma = (1,1,1,1)$ (all spokes red) is realisable with fiber count $N(\sigma ) = 2$ --- the cycle alternates between the two remaining colors in two ways. \textbf  {Middle:} the alternating configuration $\sigma = (1,2,1,2)$ is \emph  {infeasible} ($N(\sigma ) = 0$); the constraints at $v_0$ and $v_1$ force $c(e_0) = 3$, but the constraint at $v_2$ then demands $c(e_1) \in \{2, 3\} \setminus \{c(e_0)\}$ contradicting the $v_1$ constraint. \textbf  {Right:} the non-uniform configuration $\sigma = (1,1,2,2)$ has fiber count $N(\sigma ) = 1$ --- a single cycle coloring extends it.\relax }}{3}{}\protected@file@percent }
+\newlabel{fig:fiber-c4}{{2}{3}}
+\@writefile{toc}{\contentsline {paragraph}{Total count.}{3}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Distribution over $\sigma $.}{3}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{Where this leads.}{4}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Two adjacent tires $T^{\mathrm  {outer}}, T^{\mathrm  {inner}}$ in a common maximal planar $G$, sharing the cycle $\gamma $. The cycle vertex $v \in V(\gamma )$ is a dual vertex for an annular face in \emph  {both} tires. The $G'$-edges at $v$ split into: two cycle edges inside $T'^{\mathrm  {outer}}_{\mathrm  {ann}}$, two cycle edges inside $T'^{\mathrm  {inner}}_{\mathrm  {ann}}$, and (in the spoke-only case) \emph  {one shared spoke edge} $vu$ pointing across the shared boundary. That single spoke edge is simultaneously a spoke of $T'^{\mathrm  {outer}}_{f'}$ and a spoke of $T'^{\mathrm  {inner}}_{f'}$, so any global edge coloring of $G'$ assigns it a single color from both sides at once.\relax }}{5}{}\protected@file@percent }
+\newlabel{fig:nesting-compatibility}{{3}{5}}
+\gdef \@abspage@last{5}

papers/coloring_nested_tire_graphs/notes/fiber_decomposition.log

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papers/coloring_nested_tire_graphs/notes/fiber_decomposition.tex

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+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{graphicx}
+\usepackage{geometry}
+\usepackage{caption}
+\usepackage{subcaption}
+\geometry{margin=1in}
+
+\title{Fiber decomposition of edge $3$-colorings\\
+       of tire annular face connectors}
+\author{}
+\date{}
+
+\newtheorem*{prop}{Proposition}
+\newtheorem*{defn}{Definition}
+
+\begin{document}
+\maketitle
+
+\section*{Why this note exists}
+
+The menagerie note gives a closed form (Section~6) for $P_e(G, k)$ when
+$G$ is a $2$-connected outerplanar graph with $\Delta \le 3$: a polygon
+$C_n$ with a non-crossing chord matching $M$.  But the object we
+actually want to count over, the \emph{tire annular face connector}
+$T'_{f'}$ (Definition~1.16 of the main paper), is not that graph --- it
+has extra pendant edges hanging off the cycle, one per non-chord
+boundary vertex.  These pendants are the \emph{spokes}
+(Definition~1.17).
+
+This note explains the picture below: $T'_{f'}$ factors as a
+\emph{core} (the polygon-with-matching $C_n + M = T'_{\mathrm{ann}}$
+that the menagerie handles) plus a set of \emph{spoke pendants} reaching
+into the neighbouring tires.  Each edge $3$-coloring of $T'_{f'}$ is
+the data of (i) a core edge coloring and (ii) a spoke coloring, and the
+spoke coloring is what compatibility-with-neighbours acts on.
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.92\textwidth]{fig_fiber_setup.png}
+\caption{Anatomy of a tire annular face connector $T'_{f'}$ in the
+spoke-only case.  \textbf{Left:} the core $T'_{\mathrm{ann}} = C_n + M$ (here
+$\theta(1,3,3)$), the object the menagerie \S6 formula counts.
+\textbf{Right:} $T'_{f'}$ adds one pendant spoke edge for each non-chord
+boundary vertex; the spoke vertex is a dual vertex of a face of $G$
+\emph{outside} the tire annulus.  In a nested-tire setting, that
+outside face belongs to the next tire over.}
+\label{fig:fiber-setup}
+\end{figure}
+
+\section*{Setup}
+
+Let $T \subseteq G$ be a tire with annular faces $F_{\mathrm{ann}}$,
+let $T'_{\mathrm{ann}} := G'[\{d_f : f \in F_{\mathrm{ann}}\}]$ be the
+tire annular subgraph, fix a face $f'$ of $T'_{\mathrm{ann}}$ with
+boundary walk $V(f')$, and let $T'_{f'}$ be the tire annular face
+connector (Definition~1.16).  Write
+\[
+   S(f') \;:=\; V(T'_{f'}) \setminus V(f')
+   \;=\; V_{\mathrm{out}}(T'_{f'}) \sqcup V_{\mathrm{in}}(T'_{f'})
+\]
+for the set of \emph{spokes} (outer and inner).  In the spoke-only
+setting (Remark~1.16.1) each spoke $u \in S(f')$ contributes a single
+pendant edge $vu$ to $T'_{f'}$ for a unique cycle vertex
+$v = v(u) \in V(f')$.  The \emph{spoke edge set} is
+\[
+   E_S \;:=\; \{\, v(u)\,u \,:\, u \in S(f') \,\} \;\subseteq\; E(T'_{f'}),
+\]
+and $E(T'_{f'}) = E(T'_{\mathrm{ann}}) \sqcup E_S$.
+
+\section*{Spoke configurations and fibers}
+
+\begin{defn}[Spoke configuration]
+A \emph{spoke configuration} on $T'_{f'}$ (with palette $[k]$) is a map
+$\sigma : E_S \to [k]$.  Equivalently, $\sigma$ assigns a color to the
+$G'$-edge from each spoke $u \in S(f')$ to its attachment vertex
+$v(u) \in V(f')$.
+\end{defn}
+
+\begin{defn}[Fiber]
+For $\sigma : E_S \to [k]$ the \emph{fiber} of $T'_{f'}$ over $\sigma$
+is
+\[
+   N(T'_{f'};\,\sigma) \;:=\; \#\bigl\{\,\text{proper edge $k$-colorings of }
+       T'_{f'} \text{ that restrict to } \sigma \text{ on } E_S\,\bigr\}.
+\]
+\end{defn}
+
+Trivially
+\[
+   \boxed{\;P_e(T'_{f'},\,k) \;=\; \sum_{\sigma : E_S \to [k]}
+                                    N(T'_{f'};\,\sigma)\;}
+\]
+and the support of the sum is the set of \emph{realisable} spoke
+configurations.
+
+\paragraph{What the fiber count really is.}
+Fixing $\sigma$ pins one extra color at each cycle vertex $v(u)$: the
+spoke edge $v(u)u$ contributes a forbidden color
+$\sigma(v(u)u)$ at $v(u)$.  After this, the remaining problem is to
+proper-edge-color the core $T'_{\mathrm{ann}} = C_n + M$ subject to
+vertex-level forbidden colors --- precisely the constrained transfer
+matrix problem in the ``General method'' paragraph of menagerie \S6,
+with the chord-color part of the forbidden-set assignment replaced by
+the spoke-color part.
+
+If we write $C^{(v)}_\sigma \subseteq [k]$ for the set of forbidden
+colors at vertex $v$ coming from spokes (so $|C^{(v)}_\sigma|$ is the
+number of spokes attached at $v$; in the spoke-only case it is $0$ or
+$1$), then
+\[
+   N(T'_{f'};\,\sigma) \;=\; \sum_{(c_1,\dots,c_r)\in[k]^r}
+       N\!\bigl(C_n;\,\mathrm{forbidden}_{M}(c_1,\dots,c_r)
+                       \cup C^{(\cdot)}_\sigma,\,k\bigr),
+\]
+i.e.\ the menagerie sum but with the extra spoke-forbidden colors
+appended to the forbidden sets at each cycle vertex.
+
+\paragraph{Special case: $E_S = \emptyset$.}
+If there are no spokes (e.g.\ the spoke-free setting), $\sigma$ is the
+empty map and the unique fiber is
+$N(T'_{f'};\,\sigma_\emptyset) = P_e(T'_{\mathrm{ann}}, k)$, which is
+exactly the menagerie \S6 formula.
+
+\section*{Small worked example: $C_4$ with four spokes at $k=3$}
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.95\textwidth]{fig_fiber_c4_example.png}
+\caption{Three spoke configurations $\sigma$ on $C_4$ with four
+pendant spokes, with cycle vertices labelled $v_0, v_1, v_2, v_3$ and
+spoke vertices $u_0, u_1, u_2, u_3$.  \textbf{Left:} the constant
+configuration $\sigma = (1,1,1,1)$ (all spokes red) is realisable with
+fiber count $N(\sigma) = 2$ --- the cycle alternates between the two
+remaining colors in two ways.  \textbf{Middle:} the alternating
+configuration $\sigma = (1,2,1,2)$ is \emph{infeasible} ($N(\sigma) =
+0$); the constraints at $v_0$ and $v_1$ force $c(e_0) = 3$, but the
+constraint at $v_2$ then demands $c(e_1) \in \{2, 3\} \setminus
+\{c(e_0)\}$ contradicting the $v_1$ constraint.  \textbf{Right:} the
+non-uniform configuration $\sigma = (1,1,2,2)$ has fiber count
+$N(\sigma) = 1$ --- a single cycle coloring extends it.}
+\label{fig:fiber-c4}
+\end{figure}
+
+Let $T'_{f'}$ be the $4$-cycle $C_4 = v_0 v_1 v_2 v_3$ with one pendant
+spoke $v_i u_i$ at each cycle vertex.  Each $v_i$ has degree $3$, so
+proper edge $3$-coloring forces $\{c(s_i),\,c(e_{i-1}),\,c(e_i)\} =
+\{1,2,3\}$ at every $v_i$, where $s_i := v_i u_i$ and $e_i$ is the
+cycle edge from $v_i$ to $v_{i+1}$ (indices mod $4$).
+
+Fix a spoke configuration $\sigma = (\sigma_0,\sigma_1,\sigma_2,\sigma_3)$.
+At each $v_i$, the two cycle edges $e_{i-1}, e_i$ are the two colors of
+$[3] \setminus \{\sigma_i\}$ in some order.  This gives the constraints
+\[
+   c(e_i) \in [3] \setminus \{\sigma_i, \sigma_{i+1}\},
+   \qquad i = 0,1,2,3,
+\]
+together with $c(e_{i-1}) \ne c(e_i)$ at each $v_i$.
+
+\paragraph{Total count.}
+For $C_4$ alone, $P_e(C_4, 3) = P_v(C_4, 3) = 2^4 + 2 = 18$.  At $k=3$,
+each spoke is forced (the unique color avoiding its endpoint's two
+cycle-edge colors), so $P_e(T'_{f'}, 3) = 18$ as well: the pendants
+contribute a factor of~$1^4$.
+
+\paragraph{Distribution over $\sigma$.}
+The map ``edge $3$-coloring of $T'_{f'}$'' $\mapsto$ ``induced
+$\sigma$'' is surjective onto its image, and
+\[
+   \sum_{\sigma \in [3]^4} N(T'_{f'};\,\sigma) \;=\; 18.
+\]
+The fiber sizes for the three configurations in
+Figure~\ref{fig:fiber-c4} can be checked directly from the constraints
+above; doing the same for every $\sigma \in [3]^4$ would partition the
+$18$ colorings according to the spoke pattern they induce.
+
+\section*{Why this matters for nested tires}
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.95\textwidth]{fig_nesting_compatibility.png}
+\caption{Two adjacent tires $T^{\mathrm{outer}}, T^{\mathrm{inner}}$ in
+a common maximal planar $G$, sharing the cycle $\gamma$.  The cycle
+vertex $v \in V(\gamma)$ is a dual vertex for an annular face in
+\emph{both} tires.  The $G'$-edges at $v$ split into: two cycle edges
+inside $T'^{\mathrm{outer}}_{\mathrm{ann}}$, two cycle edges inside
+$T'^{\mathrm{inner}}_{\mathrm{ann}}$, and (in the spoke-only case)
+\emph{one shared spoke edge} $vu$ pointing across the shared boundary.
+That single spoke edge is simultaneously a spoke of $T'^{\mathrm{outer}}_{f'}$
+and a spoke of $T'^{\mathrm{inner}}_{f'}$, so any global edge coloring
+of $G'$ assigns it a single color from both sides at once.}
+\label{fig:nesting-compatibility}
+\end{figure}
+
+The whole point of the fiber decomposition is that, in a nested-tire
+configuration, the spoke edge set of the inner tire is (a subset of) the
+spoke edge set of the outer tire \emph{regarded from the other side}.
+Specifically, if two tires $T^{\mathrm{outer}}, T^{\mathrm{inner}}$ share a
+cycle $\gamma$ in $G$, then for $v \in V(\gamma)$ the unique
+$G'$-edge at $v$ that points out of $\gamma$'s annular faces serves as
+an outer-spoke edge of one tire's face connector and an inner-spoke
+edge of the other's (Figure~\ref{fig:nesting-compatibility}).
+
+So an edge $3$-coloring of the whole region decomposes as: a coloring
+of each tire's face connector, with the spoke configurations matched
+on the shared cycle.  Writing
+$\sigma_{\mathrm{shared}}$ for the common spoke configuration on the
+shared boundary, this is
+\[
+   \#\{\text{joint colorings on } T^{\mathrm{outer}} \cup
+       T^{\mathrm{inner}}\}
+   \;=\; \sum_{\sigma_{\mathrm{shared}}}
+         N(T'^{\mathrm{outer}}_{f'};\, \sigma_{\mathrm{shared}},\, *)
+         \cdot
+         N(T'^{\mathrm{inner}}_{f'};\, \sigma_{\mathrm{shared}},\, *)
+\]
+(with $*$ standing for the other-side spoke configuration, which is
+free).  This is the conductivity step $\phi_B$ of the chain pigeonhole
+sketch from the conversation that prompted this note: the inner tire's
+profile, projected to its outer-side spoke configurations, must
+intersect the outer tire's profile, projected to its inner-side spoke
+configurations.
+
+\paragraph{Where this leads.}
+Two natural quantitative questions:
+\begin{enumerate}
+  \item For a fixed face connector $T'_{f'}$ and a fixed boundary side,
+        what is the distribution of $|N(T'_{f'};\,\sigma)|$ over
+        $\sigma$?  How concentrated is it, and how large is the
+        realisable support?
+  \item For two nested tires to admit a joint edge $3$-coloring it
+        suffices that the supports of $\sigma$ overlap.  When can we
+        guarantee this from the cycle length alone (which controls
+        $|[k]^{|E_S|}|$ on the shared side)?
+\end{enumerate}
+The fiber form makes both questions concrete: the first is a fiber
+distribution to compute, the second is a covering / pigeonhole on $\sigma$.
+
+\end{document}