From 56649b428dbc5322ae2d473ce1803d22791c06f3 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 21:25:36 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20add=20pict?= =?UTF-8?q?ure=20for=20menagerie=20=C2=A76=20(=CE=B8(1,3,3))?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds fig_theta133.png illustrating the polygon-with-one-chord case: hexagon C_6 with the chord between v_0 and v_3 (the two trivalent vertices), edges colored with a valid proper 3-edge-coloring using the 3 colors a, b, c. Replaces the previous bracketed placeholder text in ยง6 of menagerie.tex. Adds draw_theta_133.py generator script. Paper stays at 4 pages. Co-Authored-By: Claude Opus 4.7 --- .../notes/draw_theta_133.py | 97 ++++++++++++++++++ .../notes/fig_theta133.png | Bin 0 -> 44407 bytes .../notes/menagerie.log | 64 ++++++------ .../notes/menagerie.pdf | Bin 227201 -> 262034 bytes .../notes/menagerie.tex | 3 +- 5 files changed, 133 insertions(+), 31 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/notes/draw_theta_133.py create mode 100644 papers/coloring_nested_tire_graphs/notes/fig_theta133.png diff --git a/papers/coloring_nested_tire_graphs/notes/draw_theta_133.py b/papers/coloring_nested_tire_graphs/notes/draw_theta_133.py new file mode 100644 index 0000000..b227ce9 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/draw_theta_133.py @@ -0,0 +1,97 @@ +"""Draw theta(1, 3, 3) = hexagon with one chord, illustrating section 6 +of the menagerie. Shows a proper 3-edge-coloring with the two +trivalent vertices highlighted.""" +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_theta133.png') + + # Hexagon: 6 cycle vertices in regular hexagon, with v_0 at top and v_3 at bottom + # so the chord v_0-v_3 is the vertical diameter. + n = 6 + 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)] + chord = (0, 3) + + # Proper 3-edge-coloring: + # chord = color C, e_0 = A, e_1 = C, e_2 = B, e_3 = A, e_4 = C, e_5 = B + A = '#1f77b4' # blue + B = '#d62728' # red + C = '#2ca02c' # green + + cycle_colors = [A, C, B, A, C, B] + chord_color = C + + fig, ax = plt.subplots(figsize=(4.5, 4.0)) + + # Draw 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) + + # Draw chord (slightly dashed to distinguish from polygon edges) + x1, y1 = pos[chord[0]]; x2, y2 = pos[chord[1]] + ax.plot([x1, x2], [y1, y2], color=chord_color, linewidth=3.0, + linestyle=(0, (3, 2)), solid_capstyle='round', zorder=1) + + # Draw vertices, highlighting trivalent vertices in a darker shade + trivalent = {0, 3} + for v, (x, y) in pos.items(): + if v in trivalent: + ax.plot(x, y, 'o', color='#333333', markersize=22, zorder=2) + label_color = 'white'; fontsize = 11 + else: + ax.plot(x, y, 'o', color='#888888', markersize=18, zorder=2) + label_color = 'white'; fontsize = 10 + ax.annotate(f"$v_{v}$", (x, y), color=label_color, ha='center', + va='center', fontsize=fontsize, fontweight='bold', zorder=3) + + # Annotations: indicate the 3 paths and the chord + ax.annotate('chord (path of length 1)', + xy=(0, 0), xytext=(1.65, 0.05), + fontsize=9, color=chord_color, + arrowprops=dict(arrowstyle='->', color=chord_color, lw=1.2)) + ax.annotate('right path (length 3)', + xy=(1.0, -0.05), xytext=(1.6, -0.75), + fontsize=9, color='#444', + arrowprops=dict(arrowstyle='->', color='#444', lw=1.0)) + ax.annotate('left path (length 3)', + xy=(-1.0, -0.05), xytext=(-2.45, -0.75), + fontsize=9, color='#444', + arrowprops=dict(arrowstyle='->', color='#444', lw=1.0)) + + # Legend for edge colors + legend_items = [ + plt.Line2D([], [], color=A, linewidth=3, label='color $a$'), + plt.Line2D([], [], color=B, linewidth=3, label='color $b$'), + plt.Line2D([], [], color=C, linewidth=3, label='color $c$'), + plt.Line2D([], [], marker='o', color='w', markerfacecolor='#333333', + markersize=12, + label='trivalent (degree-3) vertex'), + ] + ax.legend(handles=legend_items, loc='upper left', + bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False) + + ax.set_xlim(-2.6, 2.7) + ax.set_ylim(-1.4, 1.4) + ax.set_aspect('equal'); ax.axis('off') + ax.set_title(r'$\theta(1, 3, 3)$: hexagon $C_6$ with one chord' + '\n' + + r'(2-connected outerplanar, $\Delta = 3$, $P_e(\theta(1,3,3),3)=30$)', + 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_theta133.png b/papers/coloring_nested_tire_graphs/notes/fig_theta133.png new file mode 100644 index 0000000000000000000000000000000000000000..0e2688c07e2f87a7f2279ae40a32dc01aa6d8609 GIT binary patch literal 44407 zcmd43bx@V<_ctmEB1(t>5&{yUba#lfC`fk+NO!lQ0!oJ<4IW#igK z{Z^ceXkF-2=PSNJjT-`BN!iMRjr z&ow0P9I5~F#cO0IFY)hd={0Wu|GaWKxP||JE|9>^NB^H!I&fY^{O6StHtoL?|8vdG zzWx)^Eg`jg&#`UFCUQb+9XOyNyU4=DAH^AF)*MI!>YeIRC?mK zv)mKZ)YO#7@7QjR%2q3gKk;fjKe5c(c-MJ5RGDI}aQ?=ABB!bRJ_Qkr&g$k@SRT#b z7Iz`SyLah?JrAPexh=cld+E=wtgpZRES(#UmUyYsOHa>ZHns~r7_#2&mqgNz{$upem z*CTu5IOQ5V_Kp7xWgxxXyU+dc$TE>WgQ;TpjJMk;>s+^D21GA(p2aZbG-~r5_`Q3V zVKq@Y)a2{S?|o8!bQ$yIHB8ry$>o1HPFVBe4a((3;c$i=v1YNs+okT$a6BZx-m?e 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