From dfca45e9135c360ac089883e5fc741504c9be498 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 21:30:53 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20add=203-ch?= =?UTF-8?q?ord=20example=20calculation=20in=20menagerie=20=C2=A76?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds a worked example: G = C_8 with three non-crossing chords {(v_0,v_2), (v_3,v_7), (v_4,v_6)}. Walks through the calculation of P_e(G, 3) by propagating constraints: 1. Fix chord c_0 = a (3 choices). 2. Forces {c(e_0), c(e_7)} = {b, c} and {c(e_1), c(e_2)} = {b, c} at v_0 and v_2; cycle constraint at v_1 ties them together. 3. Propagating to chord 3-7 forces c_3 = a and the adjacent cycle edges to alternate {b, c}. 4. Propagating to chord 4-6 forces c_4 = a and cycle edges continue the alternation. Result: cycle edges alternate b, c around C_8 (OK since |C_8| is even); all 3 chords get the same color a. Total proper 3-edge- colorings: 3 (choice of a) × 2 (b/c assignment) = 6, verified by Sage's chromatic-polynomial computation on L(G). Note that the graph admits a UNIQUE proper 3-edge-coloring modulo permutation of the 3 colors -- the chord structure forces all three chords to take the "third" color absent on the polygon cycle. Adds: - draw_3chord_example.py - fig_3chord_example.png Paper grows from 4 to 5 pages. Co-Authored-By: Claude Opus 4.7 --- .../notes/draw_3chord_example.py | 86 ++++++++++++++++++ .../notes/fig_3chord_example.png | Bin 0 -> 57148 bytes .../notes/menagerie.aux | 3 +- .../notes/menagerie.log | 67 +++++++------- .../notes/menagerie.pdf | Bin 262034 -> 321355 bytes .../notes/menagerie.tex | 54 +++++++++++ 6 files changed, 178 insertions(+), 32 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/notes/draw_3chord_example.py create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_3chord_example.png diff --git a/papers/coloring_nested_tire_graphs/notes/draw_3chord_example.py b/papers/coloring_nested_tire_graphs/notes/draw_3chord_example.py new file mode 100644 index 0000000..efe9003 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/draw_3chord_example.py @@ -0,0 +1,86 @@ +"""Draw C_8 with three non-crossing chords {(0,2), (3,7), (4,6)}, and +overlay the unique proper 3-edge-coloring (modulo permutation of the 3 +colors).""" +import math +import os +import matplotlib.pyplot as plt + + +def main(): + here = os.path.dirname(os.path.abspath(__file__)) + out = os.path.join(here, 'fig_3chord_example.png') + + n = 8 + pos = {} + for i in range(n): + angle = math.pi / 2 - 2 * math.pi * i / n + pos[i] = (math.cos(angle), math.sin(angle)) + + cycle_edges = [(i, (i + 1) % n) for i in range(n)] + chords = [(0, 2), (3, 7), (4, 6)] + + # Coloring (derived in the menagerie note): + # all 3 chords = color "a" (call it color 1). + # cycle alternates "b", "c": e_0=b, e_1=c, e_2=b, e_3=c, e_4=b, e_5=c, e_6=b, e_7=c. + A = '#1f77b4' # color a (used by chords) + B = '#d62728' # color b + C = '#2ca02c' # color c + + cycle_colors = [B, C, B, C, B, C, B, C] + chord_color = A + + fig, ax = plt.subplots(figsize=(5.4, 4.6)) + + # Cycle edges + for i, (u, v) in enumerate(cycle_edges): + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=cycle_colors[i], linewidth=3.0, + solid_capstyle='round', zorder=1) + + # Chords (dashed to distinguish) + for u, v in chords: + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=chord_color, linewidth=3.0, + linestyle=(0, (3, 2)), solid_capstyle='round', zorder=1) + + # Vertices: chord endpoints (degree 3) darker; others (degree 2) lighter + chord_verts = {v for c in chords for v in c} + for v, (x, y) in pos.items(): + if v in chord_verts: + ax.plot(x, y, 'o', color='#333333', markersize=22, zorder=2) + ax.annotate(f"$v_{v}$", (x, y), color='white', ha='center', + va='center', fontsize=11, fontweight='bold', zorder=3) + else: + ax.plot(x, y, 'o', color='#888888', markersize=18, zorder=2) + ax.annotate(f"$v_{v}$", (x, y), color='white', ha='center', + va='center', fontsize=10, fontweight='bold', zorder=3) + + legend_items = [ + plt.Line2D([], [], color=A, linewidth=3, linestyle='--', + label='chord, color $a$'), + plt.Line2D([], [], color=B, linewidth=3, label='cycle, color $b$'), + plt.Line2D([], [], color=C, linewidth=3, label='cycle, color $c$'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#333333', + markersize=12, label='chord-endpoint (degree 3)'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#888888', + markersize=11, label='polygon-only vertex (degree 2)'), + ] + ax.legend(handles=legend_items, loc='upper left', + bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False) + + ax.set_xlim(-1.4, 2.4) + ax.set_ylim(-1.3, 1.3) + ax.set_aspect('equal'); ax.axis('off') + ax.set_title(r'$C_8$ with 3 non-crossing chords $\{02,\, 37,\, 46\}$' + + '\n' + + r'($P_e(G,3) = 6$; unique coloring up to permutation ' + r'of $\{a,b,c\}$)', + 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/fig_3chord_example.png b/papers/coloring_nested_tire_graphs/notes/fig_3chord_example.png new file mode 100644 index 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