From bfc68b7ec69531517faaf06f8a56144d47d1c34f Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 00:07:14 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add fiber decomposition note for T'_{f'} edge colorings MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../notes/draw_fiber_c4_example.py | 162 +++++++++ .../notes/draw_fiber_setup.py | 99 ++++++ .../notes/draw_nesting_compatibility.py | 155 ++++++++ .../notes/fiber_decomposition.aux | 14 + .../notes/fiber_decomposition.log | 331 ++++++++++++++++++ .../notes/fiber_decomposition.pdf | Bin 0 -> 399680 bytes .../notes/fiber_decomposition.tex | 239 +++++++++++++ .../notes/fig_fiber_c4_example.png | Bin 0 -> 48008 bytes .../notes/fig_fiber_setup.png | Bin 0 -> 57952 bytes .../notes/fig_nesting_compatibility.png | Bin 0 -> 154065 bytes 10 files changed, 1000 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/notes/draw_fiber_c4_example.py create mode 100644 papers/coloring_nested_tire_graphs/notes/draw_fiber_setup.py create mode 100644 papers/coloring_nested_tire_graphs/notes/draw_nesting_compatibility.py create mode 100644 papers/coloring_nested_tire_graphs/notes/fiber_decomposition.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/fiber_decomposition.log create mode 100644 papers/coloring_nested_tire_graphs/notes/fiber_decomposition.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/fiber_decomposition.tex create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_fiber_c4_example.png create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_fiber_setup.png create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_nesting_compatibility.png diff --git a/papers/coloring_nested_tire_graphs/notes/draw_fiber_c4_example.py b/papers/coloring_nested_tire_graphs/notes/draw_fiber_c4_example.py new file mode 100644 index 0000000..06540e8 --- /dev/null +++ b/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() diff --git a/papers/coloring_nested_tire_graphs/notes/draw_fiber_setup.py b/papers/coloring_nested_tire_graphs/notes/draw_fiber_setup.py new file mode 100644 index 0000000..2f00f9f --- /dev/null +++ b/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() diff --git a/papers/coloring_nested_tire_graphs/notes/draw_nesting_compatibility.py b/papers/coloring_nested_tire_graphs/notes/draw_nesting_compatibility.py new file mode 100644 index 0000000..23be156 --- /dev/null +++ b/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() diff --git a/papers/coloring_nested_tire_graphs/notes/fiber_decomposition.aux b/papers/coloring_nested_tire_graphs/notes/fiber_decomposition.aux new file mode 100644 index 0000000..5f96c21 --- /dev/null +++ b/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} diff --git a/papers/coloring_nested_tire_graphs/notes/fiber_decomposition.log b/papers/coloring_nested_tire_graphs/notes/fiber_decomposition.log new file mode 100644 index 0000000..c85ea3f --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/fiber_decomposition.log @@ -0,0 +1,331 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 00:03 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**fiber_decomposition.tex +(./fiber_decomposition.tex +LaTeX2e <2021-11-15> patch level 1 +L3 programming layer <2022-02-24> 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