From 388ab19db90b77eb4f414aa0b8144320b9bb63e5 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 04:30:48 -0400 Subject: [PATCH] coloring_nested_tire_graphs: pin nesting iso, factor seam lemma, add figure Rewrite Conjecture 1.20 (universal nesting) with the iso notion fixed to combinatorial with O preserved: rooted tree iso + plane-outerplanar iso of O on each tread + child/face correspondence, with B_out explicitly not required to match (essential for sub-tree embedding). Factor the technical core out as Conjecture 1.22 (seam realizability): for every k >= 3, exhibit a triangulated planar disk H_k with boundary a k-cycle whose BFS-from-boundary tree of treads is iso to a given T_1. Add Remark 1.23 stating that universal nesting reduces to seam realizability by excise-and-glue using the existing structural theorems. Reworked Remark 1.24 (motivation) keeps the compositional-colourability and universality bullets, and replaces the old open-questions paragraph with three concrete subproblems: a candidate apex-removal construction for the seam, 6-connectivity preservation as the relevant 4CT subproblem, and a justification of why the weaker iso notion is necessary. Add fig_seam_construction.png (and the matplotlib script that generates it) illustrating the seam construction on a 10-vertex G_1 with T_1 a chain of length 3; the script asserts BFS-from-boundary in H_5 reproduces ell_{G_1} on V(G_1) \ {S_1}, giving a verified small instance of the conjecture. Co-Authored-By: Claude Opus 4.7 --- .../experiments/draw_seam_construction.py | 195 +++++++++++++++++ .../fig_seam_construction.png | Bin 0 -> 405525 bytes papers/coloring_nested_tire_graphs/paper.aux | 13 +- papers/coloring_nested_tire_graphs/paper.log | 91 ++++---- papers/coloring_nested_tire_graphs/paper.pdf | Bin 623232 -> 927914 bytes papers/coloring_nested_tire_graphs/paper.tex | 202 ++++++++++++++---- 6 files changed, 420 insertions(+), 81 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py create mode 100644 papers/coloring_nested_tire_graphs/fig_seam_construction.png diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py b/papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py new file mode 100644 index 0000000..0747261 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py @@ -0,0 +1,195 @@ +"""Draw the seam-realizability construction for Conjecture~\\ref{conj:seam-realizability}. + +(a) Left: G_1, a stacked-ring triangulation with single-vertex level source + S_1 = {0}. Vertices coloured by BFS-from-S_1 level (0, 1, 2, 3). + The four colours visually correspond to the rooted tree of tire treads + of G_1 -- a chain T_0 -> T_1 -> T_2 with O^{(T_d)} = G_1[L_{d+1}] on + each tread. + +(b) Right: H_5, the apex-removal seam construction. We take G_1 \\ {S_1} + (octahedron-like, 9 vertices) and re-embed so the former fan-face + around S_1 becomes the outer face, with L_1 as the new outer boundary + of that disk. We then attach a triangulated annulus A_5 from L_1 to a + fresh boundary 5-cycle, partial H_5. Vertices coloured by + BFS-from-(partial H_5) level. + +Visual claim: the BFS level labels match between (a) and (b) on +V(G_1) \\ {S_1}; the new level-0 ring of (b) is the boundary cycle +partial H_5 of length k = 5, replacing the single-vertex source S_1 of (a). +Hence T(H_5, partial H_5) is iso (combinatorial, O-preserved) to T(G_1, S_1). +""" +import math +import os +import networkx as nx +import matplotlib.pyplot as plt +from matplotlib.lines import Line2D + +OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__))) + +# --------------------------------------------------------------------------- +# (a) Build G_1: source 0 + concentric triangular rings L_1, L_2, L_3. +# --------------------------------------------------------------------------- +RINGS = 3 +pos_G = {0: (0.0, 0.0)} +ring_G = {0: [0]} +nxt = 1 +for r in range(1, RINGS + 1): + ids = [] + for j in range(3): + ang = math.radians(90 + 120 * j) + pos_G[nxt] = (r * math.cos(ang), r * math.sin(ang)) + ids.append(nxt) + nxt += 1 + ring_G[r] = ids + +G1 = nx.Graph() +G1.add_nodes_from(pos_G) +for v in ring_G[1]: + G1.add_edge(0, v) +for r in range(1, RINGS + 1): + a, b, c = ring_G[r] + G1.add_edges_from([(a, b), (b, c), (c, a)]) +for r in range(1, RINGS): + inner, outer = ring_G[r], ring_G[r + 1] + for j in range(3): + G1.add_edge(inner[j], outer[j]) # spoke + G1.add_edge(inner[j], outer[(j + 1) % 3]) # annulus diagonal + +assert G1.number_of_edges() == 3 * G1.number_of_nodes() - 6, "G_1 not a triangulation" + +level_G = nx.shortest_path_length(G1, source=0) + +# --------------------------------------------------------------------------- +# (b) Build H_5: re-embed (G_1 \ {S_1}) so the former S_1 fan-face is the +# outer face (L_1 becomes the outer-most ring of that disk), then attach +# a triangulated annulus A_5 to a fresh 5-cycle partial H_5. +# --------------------------------------------------------------------------- +# Concentric placement: L_3 innermost (was G_1's outer face), L_1 outermost +# among G_1-derived vertices, partial H_5 outermost overall. +pos_H = {} +# Concentric placement with the SAME angles as panel (a): L_3 innermost, +# L_1 outermost among G_1-derived vertices. This is the topological flip +# (re-embedding) of (a); same vertex labels, same edges (minus S_1), but +# the ring with the smallest G_1-level is now at the largest radius. +RING_R = {1: 2.4, 2: 1.6, 3: 0.8} +for r in [1, 2, 3]: + for j, v in enumerate(ring_G[r]): + ang = math.radians(90 + 120 * j) + pos_H[v] = (RING_R[r] * math.cos(ang), RING_R[r] * math.sin(ang)) + +# partial H_5 vertices u0..u4 on a regular pentagon +BOUNDARY = [f'u{j}' for j in range(5)] +R_BDY = 3.6 +for j, u in enumerate(BOUNDARY): + ang = math.radians(90 + 72 * j) + pos_H[u] = (R_BDY * math.cos(ang), R_BDY * math.sin(ang)) + +H = nx.Graph() +H.add_nodes_from(pos_H) +# inherit G_1 \ {S_1} edges, but recompute embedding rotations are positional +for u, v in G1.edges(): + if 0 in (u, v): + continue + H.add_edge(u, v) +# partial H_5 cycle (the new outer boundary) +boundary_edges = [(BOUNDARY[j], BOUNDARY[(j + 1) % 5]) for j in range(5)] +H.add_edges_from(boundary_edges) +# annular edges A_5: a, b, c (= ring_G[1]) match to consecutive u_i ranges +a, b, c = ring_G[1] +annular_pairs = [ + (a, 'u0'), (a, 'u1'), (a, 'u2'), + (b, 'u2'), (b, 'u3'), (b, 'u4'), + (c, 'u4'), (c, 'u0'), +] +H.add_edges_from(annular_pairs) + +# H is a triangulated planar disk: all bounded faces triangles, outer face +# the 5-cycle partial H_5. Verify edge count by Euler with f outer = 5-gon: +# t bounded triangles, F = t + 1, V - E + F = 2 => E = V + t - 1; +# 3t + 5 = 2E => t = 2V - 7. Here V = 14, so t = 21, E = 34. +assert H.number_of_edges() == 34, f"unexpected edge count {H.number_of_edges()}" + +level_H = nx.multi_source_dijkstra_path_length(H, set(BOUNDARY)) + +# Verify the seam claim: BFS levels on V(G_1) \ {S_1} match between G_1 and H. +for v in G1.nodes(): + if v == 0: + continue + assert level_H[v] == level_G[v], ( + f"level mismatch at v={v}: G_1 level {level_G[v]}, H level {level_H[v]}" + ) + +# --------------------------------------------------------------------------- +# Draw. +# --------------------------------------------------------------------------- +LEVEL_COLOR = {0: '#1e293b', 1: '#475569', 2: '#94a3b8', 3: '#cbd5e1'} + +fig, axes = plt.subplots(1, 2, figsize=(16, 8.5)) + + +def draw_panel(ax, graph, pos, levels, title, *, boundary=None, annular=None): + nx.draw_networkx_edges(graph, pos, ax=ax, edge_color='#d1d5db', width=1.2) + if annular: + nx.draw_networkx_edges(graph, pos, edgelist=annular, ax=ax, + edge_color='#f59e0b', width=1.8) + if boundary: + nx.draw_networkx_edges(graph, pos, edgelist=boundary, ax=ax, + edge_color='#dc2626', width=2.4) + for v, (x, y) in pos.items(): + lev = levels[v] + ax.scatter([x], [y], s=560, color=LEVEL_COLOR[lev], + edgecolors='black', linewidths=1.0, zorder=3) + ax.text(x, y, f'{v}\n$\\ell{{=}}{lev}$', ha='center', va='center', + color='white', fontsize=8.5, fontweight='bold', zorder=4) + ax.set_aspect('equal') + ax.axis('off') + ax.set_title(title, fontsize=11) + + +draw_panel( + axes[0], G1, pos_G, level_G, + r"$(a)$ $G_1$ with single-vertex source $S_1 = \{0\}$." "\n" + r"BFS from $S_1$ gives levels $\ell = 0, 1, 2, 3$ " + r"(rings $\{0\}, L_1, L_2, L_3$).", +) + +draw_panel( + axes[1], H, pos_H, level_H, + r"$(b)$ $H_5$: apex-removal seam." "\n" + r"Outer 5-cycle $\partial H_5$ (red) replaces $S_1$; " + r"annulus $A_5$ (orange) glues $\partial H_5$ to $L_1$.", + boundary=boundary_edges, annular=annular_pairs, +) + +legend = [ + Line2D([0], [0], marker='o', color='w', label=r'level $\ell = 0$', + markerfacecolor=LEVEL_COLOR[0], markeredgecolor='black', markersize=12), + Line2D([0], [0], marker='o', color='w', label=r'level $\ell = 1$ ($L_1$)', + markerfacecolor=LEVEL_COLOR[1], markeredgecolor='black', markersize=12), + Line2D([0], [0], marker='o', color='w', label=r'level $\ell = 2$ ($L_2$)', + markerfacecolor=LEVEL_COLOR[2], markeredgecolor='black', markersize=12), + Line2D([0], [0], marker='o', color='w', label=r'level $\ell = 3$ ($L_3$)', + markerfacecolor=LEVEL_COLOR[3], markeredgecolor='black', markersize=12), + Line2D([0], [0], color='#dc2626', lw=2.4, label=r'$\partial H_5$ (boundary $k$-cycle, $k=5$)'), + Line2D([0], [0], color='#f59e0b', lw=1.8, label=r'annulus $A_5$ edges'), +] +fig.legend(handles=legend, loc='lower center', ncol=3, fontsize=10, framealpha=0.95) + +fig.suptitle( + r"Seam realizability (Conjecture 1.22): " + r"$\mathcal{T}(H_5, \partial H_5) \cong \mathcal{T}(G_1, S_1)$ " + r"as rooted trees of tire treads (combinatorial, $O$-preserved). " + r"BFS distances from $\partial H_5$ in $H_5$ reproduce $\ell_{G_1}$ " + r"on $V(G_1) \setminus \{S_1\}$.", fontsize=12, +) +fig.tight_layout(rect=[0, 0.08, 1, 0.95]) + +out = os.path.join(OUT_DIR, 'fig_seam_construction.png') +fig.savefig(out, dpi=180, bbox_inches='tight') +plt.close(fig) + +print(f'G_1: |V|={G1.number_of_nodes()}, |E|={G1.number_of_edges()}, ' + f'levels: {sorted(set(level_G.values()))}') +print(f'H_5: |V|={H.number_of_nodes()}, |E|={H.number_of_edges()}, ' + f'levels: {sorted(set(level_H.values()))}') +print(f'wrote {out}') diff --git a/papers/coloring_nested_tire_graphs/fig_seam_construction.png b/papers/coloring_nested_tire_graphs/fig_seam_construction.png new file mode 100644 index 0000000000000000000000000000000000000000..a80da8e4bbc39dbd520ebcb4e9d8525f6fbe23b0 GIT binary patch literal 405525 zcmdSBhgVZuyElq`i(;V)27#SECn6&Px z88I+0{$yY{vHsUdaO6GX?PlWo2ft8&Ofdy*eZU1%Xii*I`qyIW~ z`tK(k|2lNxN_|NB#u)m`uX*Hyu%BMxaU&ixrqx$^E~eKM|} 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