From 4810121ec43e4c85f9dae88837af6cb540a54fcf Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 23:21:50 -0400 Subject: [PATCH] coloring_nested_tire_graphs: cycle sources, tire-tree decomposition, seam lemma Definition 1.1 (Level source) is broadened: a level source is now a set that is either a single vertex or a simple cycle, splitting the old notion into 'vertex source' and 'cycle source'. Downstream theorems (Prop 1.7, Lemma 1.8, Thm 1.17) remain stated for vertex sources but are referenced by the new material with cycle sources. New Theorem 1.19 (Tire-tree decomposition): for any tread T in T(G, {v_0}) at depth d >= 1 with outer cycle C_T, the sub-graph G_T inside C_T on the side away from v_0 is a triangulated disk; taking C_T as a cycle source, T(G_T, C_T) is canonically iso to the sub-tree of T(G, {v_0}) rooted at T. Proof in three steps: (D1) triangulated-disk via Jordan curve, (D2) level-shift ell_{G_T}(.) = ell_G(.) - d via shortest-path stays in R_T, (D3) component-of-G'_k bijection with descendants of T. Figure fig_tire_tree_decomposition.png (and its generator experiments/draw_tire_tree_decomposition.py) illustrates the decomposition on a 13-vertex, 5-level example with four nested seams C_{T_R}, C_{T_L}, C_{T_{LL}}, C_{T_{LLL}}; the generator script verifies the level-shift assertion on this instance. Vertex positions are hand-tuned in TikZiT and copied back; the right-panel labels are rotated relative to the parent G to emphasise the new role of C_{T_L} as cycle source. New Definition 1.21 (Seam): a seam is the outer-boundary cycle B_out^{(T)} of a non-root tread T, separating G into the seam interior G_T and seam exterior G_C^{ext}. Notation Col(X | C) for boundary-restricted 4-colourings is also defined here. New Definition 1.22 (Partial tire tree): G_{T_r}^{circle} = G_{T_r} with V(C_{T_r}) removed, i.e. the strict interior of the triangulated disk inside the seam. New Lemma 1.23 (Seam edges shared by <= one other depth-d seam): an edge on the seam of a depth-d tread T is in the seam of at most one other depth-d tread T'. Proof via inner-dual-of-outerplanar- is-a-tree: C_T bounds a face of the parent's O^{(T_p)} (outerplanar), so each edge of O^{(T_p)} lies in at most two of its bounded face cycles, giving at most one sibling seam containing e. New Conjecture 1.24 (Seam structure of minimum 4CT counterexamples, sketch): a hypothetical minimum 4CT counterexample has bilateral colourability, bilateral incompatibility, Birkhoff's seam-length >= 6 bound, and an innermost obstruction at a leaf tread T^* whose seam interior is one of a finite list of minimal seam configurations, with the boundary palette restriction propagating outward along the root-to-T^* obstruction chain. Co-Authored-By: Claude Opus 4.7 --- .../draw_tire_tree_decomposition.py | 480 ++++++++++++++++++ .../fig_tire_tree_decomposition.png | Bin 0 -> 372371 bytes papers/coloring_nested_tire_graphs/paper.aux | 13 +- papers/coloring_nested_tire_graphs/paper.log | 96 ++-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 618557 -> 927226 bytes papers/coloring_nested_tire_graphs/paper.tex | 318 +++++++++++- 6 files changed, 863 insertions(+), 44 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py create mode 100644 papers/coloring_nested_tire_graphs/fig_tire_tree_decomposition.png diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py b/papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py new file mode 100644 index 0000000..cd4740b --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py @@ -0,0 +1,480 @@ +"""Draw the tire-tree decomposition for Theorem~\\ref{thm:tire-tree-decomposition}. + +Uses a Tutte embedding (barycentric solve with the outer face fixed on a +convex polygon) to give a planar straight-line layout. + +Panel (a): G with 13 vertices, 5 levels (v_0 at level 0; L_1, L_2, L_3, L_4 +on subsequent levels). Tutte outer face = h-g_1-g_2 (the deepest face from +v_0), so v_0 ends up roughly central and the outer triangle is "outermost" +in BFS-from-v_0 distance. Tree-of-treads has 5 nodes: + T_0 + / \\ + T_R T_L + | + T_LL + | + T_LLL + +Panel (b): G_{T_L} on 10 vertices, with Tutte outer face = C_{T_L} = {a,c,d}. +Levels in G_{T_L} satisfy ell_{G_{T_L}}(.) = ell_G(.) - 1 on V(G_{T_L}); +T(G_{T_L}, C_{T_L}) is the chain T_L -> T_LL -> T_LLL. +""" +import math +import os +import networkx as nx +import numpy as np +import matplotlib.pyplot as plt +from matplotlib.lines import Line2D + +OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__))) + +# --------------------------------------------------------------------------- +# Build G. +# --------------------------------------------------------------------------- +G = nx.Graph() +G.add_nodes_from(['v0', 'a', 'b', 'c', 'd', 'e', + 'f1', 'f2', 'f3', 'g1', 'g2', 'g3', 'h']) + +# v_0 fan +for v in ['a', 'b', 'c', 'd']: + G.add_edge('v0', v) +# L_1 4-cycle +G.add_edges_from([('a', 'b'), ('b', 'c'), ('c', 'd'), ('d', 'a')]) +# Chord a-c (outside L_1 in the planar embedding) +G.add_edge('a', 'c') +# e-fan (Region R) +for v in ['a', 'b', 'c']: + G.add_edge('e', v) +# f-triangle (Region L) +G.add_edges_from([('f1', 'f2'), ('f2', 'f3'), ('f3', 'f1')]) +# Annular L_1 -> f-triangle: f_1~{a,d}, f_2~{d,c}, f_3~{a,c} +G.add_edges_from([ + ('f1', 'a'), ('f1', 'd'), + ('f2', 'd'), ('f2', 'c'), + ('f3', 'a'), ('f3', 'c'), +]) +# g-triangle (inside f-triangle) +G.add_edges_from([('g1', 'g2'), ('g2', 'g3'), ('g3', 'g1')]) +# Annular f-triangle -> g-triangle: f_1~{g_1, g_3}, f_2~{g_2, g_3}, f_3~{g_1, g_2} +G.add_edges_from([ + ('f1', 'g1'), ('f1', 'g3'), + ('f2', 'g2'), ('f2', 'g3'), + ('f3', 'g1'), ('f3', 'g2'), +]) +# h-fan (inside g-triangle) +for v in ['g1', 'g2', 'g3']: + G.add_edge('h', v) + +assert G.number_of_edges() == 3 * G.number_of_nodes() - 6, ( + f"G not a triangulation: |E|={G.number_of_edges()}, " + f"expected {3 * G.number_of_nodes() - 6}" +) + +level_G = nx.shortest_path_length(G, source='v0') + +# --------------------------------------------------------------------------- +# Tutte embedding. +# --------------------------------------------------------------------------- +def tutte_embedding(graph, outer_face_cyclic, outer_radius=3.0, + angle_offset_deg=90.0, outer_weight=1.0): + """Compute (weighted) Tutte embedding. + + outer_face_cyclic: list of vertices on the outer face, in cyclic order. + Outer vertices are fixed on a regular polygon of given radius; every + interior vertex is placed at a convex combination of its neighbours' + positions (solved as a linear system). Edges incident to an outer + vertex carry weight `outer_weight`; interior-interior edges carry + weight 1. All weights positive, so Tutte's theorem still gives a + planar straight-line embedding. Larger `outer_weight` pulls interior + vertices more strongly toward the outer triangle, spreading them out. + """ + n_outer = len(outer_face_cyclic) + outer_positions = {} + for i, v in enumerate(outer_face_cyclic): + angle = math.radians(angle_offset_deg + i * 360 / n_outer) + outer_positions[v] = (outer_radius * math.cos(angle), + outer_radius * math.sin(angle)) + + outer_set = set(outer_face_cyclic) + interior = [v for v in graph.nodes() if v not in outer_set] + interior_idx = {v: i for i, v in enumerate(interior)} + m = len(interior) + if m == 0: + return outer_positions + + A = np.zeros((m, m)) + bx = np.zeros(m) + by = np.zeros(m) + + for v in interior: + i = interior_idx[v] + for u in graph.neighbors(v): + w = outer_weight if u in outer_set else 1.0 + A[i, i] += w + if u in outer_set: + bx[i] += w * outer_positions[u][0] + by[i] += w * outer_positions[u][1] + else: + j = interior_idx[u] + A[i, j] -= w + + x = np.linalg.solve(A, bx) + y = np.linalg.solve(A, by) + + pos = dict(outer_positions) + for v in interior: + i = interior_idx[v] + pos[v] = (float(x[i]), float(y[i])) + return pos + + +def stretch_radially(pos, outer_face_cyclic, alpha=0.6): + """Stretch interior positions radially outward from the outer-face centroid. + + Outer vertices are unchanged; for each interior vertex at radius r from + the outer-face centroid, replace r by r' = R^(1-alpha) * r^alpha, where + R is the radius of the outer vertices. With alpha < 1 this is a concave + stretch that pushes small radii outward. Angles are preserved, so + planarity of the underlying Tutte embedding is preserved. + """ + outer_set = set(outer_face_cyclic) + outer_positions = [pos[v] for v in outer_face_cyclic] + Cx = sum(p[0] for p in outer_positions) / len(outer_positions) + Cy = sum(p[1] for p in outer_positions) / len(outer_positions) + R = max(math.sqrt((p[0] - Cx)**2 + (p[1] - Cy)**2) + for p in outer_positions) + + new_pos = {} + for v, p in pos.items(): + if v in outer_set: + new_pos[v] = p + continue + dx, dy = p[0] - Cx, p[1] - Cy + r = math.sqrt(dx * dx + dy * dy) + if r < 1e-10: + new_pos[v] = (Cx, Cy) + continue + r_new = (R ** (1 - alpha)) * (r ** alpha) + scale = r_new / r + new_pos[v] = (Cx + dx * scale, Cy + dy * scale) + return new_pos + + +# Panel (a) positions: hand-tuned in TikZiT and copied back here so the +# matplotlib panel matches the .tikz layout the user is editing on Desktop. +# The chord a-c is drawable as a straight line at these positions; the +# d-a edge passes through f_1, so it's drawn as a slight Bezier curve. +pos_G = { + 'v0': ( 0.00, 6.0), + 'a': (-5.50, -5.0), + 'b': (-2.75, 3.0), + 'c': ( 1.75, 3.0), + 'd': ( 8.00, -5.0), + 'e': (-1.50, 1.5), + 'f1': ( 1.50, -5.0), + 'f2': ( 4.00, -1.0), + 'f3': (-0.75, -1.0), + 'g1': ( 0.70, -3.0), + 'g2': ( 1.45, -1.5), + 'g3': ( 2.30, -3.0), + 'h': ( 1.55, -2.5), +} + +# --------------------------------------------------------------------------- +# G_{T_L}: subgraph inside C_{T_L} = {a, c, d}. +# --------------------------------------------------------------------------- +C_TL = {'a', 'c', 'd'} +V_GTL = C_TL | {'f1', 'f2', 'f3', 'g1', 'g2', 'g3', 'h'} +G_TL = G.subgraph(V_GTL).copy() +level_GTL = nx.multi_source_dijkstra_path_length(G_TL, C_TL) + +for v in V_GTL: + assert level_GTL[v] == level_G[v] - 1 + +# Tutte for G_{T_L}: outer face = a-d-c (= C_{T_L}, the seam itself). +# Cyclic order matches the planar boundary: a -> d -> c. +outer_face_GTL = ['a', 'd', 'c'] +pos_GTL = tutte_embedding(G_TL, outer_face_GTL, outer_radius=2.8, + angle_offset_deg=90, outer_weight=3.0) +pos_GTL = stretch_radially(pos_GTL, outer_face_GTL, alpha=0.6) + +# --------------------------------------------------------------------------- +# Edge sets. +# --------------------------------------------------------------------------- +C_TR_edges = [('a', 'b'), ('b', 'c'), ('a', 'c')] +C_TL_edges = [('a', 'd'), ('d', 'c'), ('a', 'c')] +C_TLL_edges = [('f1', 'f2'), ('f2', 'f3'), ('f3', 'f1')] +C_TLLL_edges = [('g1', 'g2'), ('g2', 'g3'), ('g3', 'g1')] +fan_edges = [('v0', v) for v in ['a', 'b', 'c', 'd']] + +# --------------------------------------------------------------------------- +# Tree inset helper. +# --------------------------------------------------------------------------- +def draw_tree_inset(parent_ax, position, tree_nodes, tree_edges, node_pos, + label_text, highlight=None, title=None, + xlim=(-1.4, 1.4), ylim=(-3.6, 0.6)): + ax = parent_ax.inset_axes(position) + for u, v in tree_edges: + x0, y0 = node_pos[u] + x1, y1 = node_pos[v] + ax.plot([x0, x1], [y0, y1], color='black', lw=1.4, zorder=1) + for n in tree_nodes: + x, y = node_pos[n] + is_hi = highlight and n in highlight + ax.scatter([x], [y], s=380, color='#fde68a' if is_hi else '#e5e7eb', + edgecolors='black', linewidths=1.8 if is_hi else 1.0, zorder=3) + ax.text(x, y, label_text[n], ha='center', va='center', + fontsize=7.5, fontweight='bold', zorder=4) + ax.set_xlim(*xlim); ax.set_ylim(*ylim) + ax.set_aspect('equal') + ax.set_xticks([]); ax.set_yticks([]) + if title: + ax.set_title(title, fontsize=8.5, pad=2) + for spine in ax.spines.values(): + spine.set_edgecolor('#9ca3af') + spine.set_linewidth(0.8) + ax.patch.set_facecolor('#f9fafb') + + +# --------------------------------------------------------------------------- +# Draw. +# --------------------------------------------------------------------------- +LEVEL_COLOR = { + 0: '#1e293b', + 1: '#475569', + 2: '#94a3b8', + 3: '#cbd5e1', + 4: '#f1f5f9', +} + +fig, axes = plt.subplots(1, 2, figsize=(16, 9)) + + +def _vertex_label(name): + """Render 'f1' as 'f_1' (subscript) and 'v0' as 'v_0' for display.""" + if len(name) >= 2 and name[0].isalpha() and name[1:].isdigit(): + return f'{name[0]}_{{{name[1:]}}}' + return name + + +def draw_panel(ax, graph, pos, levels, seams, fan, title, xlim, ylim, + text_color_threshold=4, label_map=None): + """seams: list of (edges, color, width) tuples; fan: edges list or None. + label_map: optional dict mapping vertex names to displayed names (so the + panel can show a relabelled version of the graph).""" + nx.draw_networkx_edges(graph, pos, ax=ax, edge_color='#d1d5db', width=1.1) + if fan is not None: + nx.draw_networkx_edges(graph, pos, edgelist=fan, ax=ax, + edge_color='#3b82f6', width=1.5, style='dashed') + for edges, color, width in seams: + nx.draw_networkx_edges(graph, pos, edgelist=edges, ax=ax, + edge_color=color, width=width) + for v, (x, y) in pos.items(): + lev = levels[v] + text_color = 'white' if lev < text_color_threshold else 'black' + display_name = (label_map or {}).get(v, v) + ax.scatter([x], [y], s=520, color=LEVEL_COLOR[lev], + edgecolors='black', linewidths=1.0, zorder=3) + ax.text(x, y, f'${_vertex_label(display_name)}$\n$\\ell{{=}}{lev}$', + ha='center', va='center', + color=text_color, fontsize=7, fontweight='bold', zorder=4) + ax.set_aspect('equal') + ax.axis('off') + ax.set_xlim(*xlim); ax.set_ylim(*ylim) + ax.set_title(title, fontsize=10.5, pad=5) + + +# Compute panel limits from the Tutte positions, with padding. +def compute_limits(pos, pad=0.5): + xs = [p[0] for p in pos.values()] + ys = [p[1] for p in pos.values()] + return (min(xs) - pad, max(xs) + pad), (min(ys) - pad, max(ys) + pad) + + +xlim_a, ylim_a = compute_limits(pos_G, pad=0.6) +xlim_b, ylim_b = compute_limits(pos_GTL, pad=0.6) + +# ============================================================================ +# Panel (a): G -- positions match Desktop/tire_tree_decomposition.tikz. +# Draw all non-seam edges as plain gray (no separate dashed fan style here, +# to match the TikZiT version). Edge a-d at y = -5 would pass through f_1 +# if drawn straight, so we render it (and seam C_{T_L}'s a-d component) as +# a slight Bezier curve via FancyArrowPatch. +# ============================================================================ +ax = axes[0] + +# Non-seam edges as straight gray lines; a-d and v_0-a are drawn curved +# below. +_curved_pairs = [{'a', 'd'}, {'v0', 'a'}] +non_seam_edges = [e for e in G.edges + if set(e) not in _curved_pairs + and set(e) not in (set(s) for s in C_TR_edges + C_TL_edges + + C_TLL_edges + C_TLLL_edges)] +nx.draw_networkx_edges(G, pos_G, edgelist=non_seam_edges, ax=ax, + edge_color='#d1d5db', width=1.1) + +# Curved edges (FancyArrowPatch). a-d dodges f_1; v_0-a is bent for +# visual balance (mirrors the TikZiT `bend right`). +from matplotlib.patches import FancyArrowPatch +def add_curved(ax, p_from, p_to, *, color, lw, rad, ls='-', zorder=2): + ax.add_patch(FancyArrowPatch( + posA=p_from, posB=p_to, + arrowstyle='-', connectionstyle=f'arc3,rad={rad}', + color=color, linewidth=lw, linestyle=ls, zorder=zorder, + )) +add_curved(ax, pos_G['a'], pos_G['d'], color='#d1d5db', lw=1.1, rad=0.15) +add_curved(ax, pos_G['v0'], pos_G['a'], color='#d1d5db', lw=1.1, rad=0.25) + +# Seams (straight, since the layout is now planar with straight chord). +# Order: bottom first (C_{T_LL}, C_{T_LLL}), then chord/L_1 seams on top. +nx.draw_networkx_edges(G, pos_G, + edgelist=[('f1', 'f2'), ('f2', 'f3'), ('f3', 'f1')], + ax=ax, edge_color='#9333ea', width=2.2) +nx.draw_networkx_edges(G, pos_G, + edgelist=[('g1', 'g2'), ('g2', 'g3'), ('g3', 'g1')], + ax=ax, edge_color='#0d9488', width=2.0) +nx.draw_networkx_edges(G, pos_G, + edgelist=[('a', 'b'), ('b', 'c')], + ax=ax, edge_color='#ea580c', width=2.4) +nx.draw_networkx_edges(G, pos_G, + edgelist=[('a', 'c'), ('d', 'c')], + ax=ax, edge_color='#dc2626', width=2.8) +add_curved(ax, pos_G['a'], pos_G['d'], color='#dc2626', lw=2.8, rad=0.15, zorder=4) + +# Vertices. +for v, (x, y) in pos_G.items(): + lev = level_G[v] + text_color = 'white' if lev < 4 else 'black' + ax.scatter([x], [y], s=520, color=LEVEL_COLOR[lev], + edgecolors='black', linewidths=1.0, zorder=5) + ax.text(x, y, f'${_vertex_label(v)}$\n$\\ell{{=}}{lev}$', + ha='center', va='center', + color=text_color, fontsize=7, fontweight='bold', zorder=6) + +ax.set_aspect('equal') +ax.axis('off') +# The a-d arc bulges below y=-5 by ~ rad * |a - d| / 2 ≈ 1.0 with rad=0.15; +# pad the y range so it isn't clipped. +xlim_a, ylim_a = compute_limits(pos_G, pad=0.8) +ylim_a = (ylim_a[0] - 1.2, ylim_a[1]) +ax.set_xlim(*xlim_a); ax.set_ylim(*ylim_a) +ax.set_title( + r"$(a)$ $G$: 13 vertices, BFS levels $\ell_G \in \{0,1,2,3,4\}$," + "\n" + r"four nested seams $C_{T_R}, C_{T_L}, C_{T_{LL}}, C_{T_{LLL}}$ " + r"(chord $a$-$c$ drawn straight in this layout).", + fontsize=10.5, pad=5, +) + +tree_pos_a = { + 'T_0': (0.0, 0.0), + 'T_R': (-0.7, -1.0), + 'T_L': (0.7, -1.0), + 'T_LL': (0.7, -2.0), + 'T_LLL': (0.7, -3.0), +} +tree_labels = { + 'T_0': r'$T_0$', 'T_R': r'$T_R$', 'T_L': r'$T_L$', + 'T_LL': r'$T_{LL}$', 'T_LLL': r'$T_{LLL}$', +} +draw_tree_inset( + axes[0], [0.70, 0.42, 0.28, 0.55], + tree_nodes=['T_0', 'T_R', 'T_L', 'T_LL', 'T_LLL'], + tree_edges=[('T_0', 'T_R'), ('T_0', 'T_L'), + ('T_L', 'T_LL'), ('T_LL', 'T_LLL')], + node_pos=tree_pos_a, label_text=tree_labels, + highlight={'T_L', 'T_LL', 'T_LLL'}, + title=r'$\mathcal{T}(G, \{v_0\})$', + xlim=(-1.4, 1.4), ylim=(-3.6, 0.6), +) + +# ============================================================================ +# Panel (b): G_{T_L}. Vertex names are rotated relative to the parent G +# (a->c, c->d, d->a; f_1->f_3, f_2->f_1, f_3->f_2; g_1->g_2, g_2->g_3, +# g_3->g_1; h stays h) -- the relabelling emphasises the new role of +# C_{T_L} as cycle source, with the boundary now naturally read as +# {a, c, d} in cyclic order from a position the user picked. +# ============================================================================ +GTL_LABEL_MAP = { + 'a': 'c', 'c': 'd', 'd': 'a', + 'f1': 'f3', 'f2': 'f1', 'f3': 'f2', + 'g1': 'g2', 'g2': 'g3', 'g3': 'g1', + 'h': 'h', +} + +draw_panel( + axes[1], G_TL, pos_GTL, level_GTL, + seams=[ + (C_TL_edges, '#dc2626', 2.8), + (C_TLL_edges, '#9333ea', 2.2), + (C_TLLL_edges, '#0d9488', 2.0), + ], + fan=None, + title=( + r"$(b)$ $G_{T_L}$ (Tutte embedding, outer face $C_{T_L} = \{a,c,d\}$):" + " 10 vertices, cycle source $C_{T_L}$,\n" + r"$\ell_{G_{T_L}}(\cdot) = \ell_G(\cdot) - 1$; " + r"$\mathcal{T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$." + ), + xlim=xlim_b, ylim=ylim_b, + text_color_threshold=3, + label_map=GTL_LABEL_MAP, +) + +sub_tree_pos = { + 'T_L': (0.0, 0.0), + 'T_LL': (0.0, -1.0), + 'T_LLL': (0.0, -2.0), +} +draw_tree_inset( + axes[1], [0.70, 0.42, 0.28, 0.55], + tree_nodes=['T_L', 'T_LL', 'T_LLL'], + tree_edges=[('T_L', 'T_LL'), ('T_LL', 'T_LLL')], + node_pos=sub_tree_pos, label_text=tree_labels, + highlight={'T_L', 'T_LL', 'T_LLL'}, + title=r'$\mathcal{T}(G_{T_L}, C_{T_L})$', + xlim=(-1.0, 1.0), ylim=(-2.6, 0.4), +) + +# ============================================================================ +# Legend + suptitle +# ============================================================================ +legend = [ + Line2D([0], [0], marker='o', color='w', label=r'$\ell = 0$', + markerfacecolor=LEVEL_COLOR[0], markeredgecolor='black', markersize=10), + Line2D([0], [0], marker='o', color='w', label=r'$\ell = 1$', + markerfacecolor=LEVEL_COLOR[1], markeredgecolor='black', markersize=10), + Line2D([0], [0], marker='o', color='w', label=r'$\ell = 2$', + markerfacecolor=LEVEL_COLOR[2], markeredgecolor='black', markersize=10), + Line2D([0], [0], marker='o', color='w', label=r'$\ell = 3$', + markerfacecolor=LEVEL_COLOR[3], markeredgecolor='black', markersize=10), + Line2D([0], [0], marker='o', color='w', label=r'$\ell = 4$', + markerfacecolor=LEVEL_COLOR[4], markeredgecolor='black', markersize=10), + Line2D([0], [0], color='#ea580c', lw=2.4, label=r'seam $C_{T_R}$'), + Line2D([0], [0], color='#dc2626', lw=2.8, label=r'seam $C_{T_L}$'), + Line2D([0], [0], color='#9333ea', lw=2.2, label=r'seam $C_{T_{LL}}$'), + Line2D([0], [0], color='#0d9488', lw=2.0, label=r'seam $C_{T_{LLL}}$'), +] +fig.legend(handles=legend, loc='lower center', ncol=5, fontsize=9.5, + framealpha=0.95, columnspacing=1.8) + +fig.suptitle( + r"Tire-tree decomposition (Theorem 1.19): every tread $T$ is the root " + r"of its own tree of tire treads $\mathcal{T}(G_T, C_T)$.", + fontsize=12, y=0.96, +) +fig.subplots_adjust(top=0.88, bottom=0.16, left=0.02, right=0.98, wspace=0.05) + +out = os.path.join(OUT_DIR, 'fig_tire_tree_decomposition.png') +fig.savefig(out, dpi=180, bbox_inches='tight') +plt.close(fig) + +print(f'G: |V|={G.number_of_nodes()}, |E|={G.number_of_edges()}, ' + f'levels {sorted(set(level_G.values()))}') +print(f'G_TL: |V|={G_TL.number_of_nodes()}, |E|={G_TL.number_of_edges()}, ' + f'levels {sorted(set(level_GTL.values()))}') +print(f'verified level shift (D2): ell_GTL(v) = ell_G(v) - 1 for all ' + f'{len(V_GTL)} v in V(G_TL)') +print(f'panel (a): hand-tuned positions (see Desktop/tire_tree_decomposition.tikz)') +print(f'Tutte (G_TL): outer face = {outer_face_GTL}') +print(f'wrote {out}') diff --git a/papers/coloring_nested_tire_graphs/fig_tire_tree_decomposition.png b/papers/coloring_nested_tire_graphs/fig_tire_tree_decomposition.png new file mode 100644 index 0000000000000000000000000000000000000000..09152d0325a61f6e273d7a97bb910d95ff615bfa GIT binary patch literal 372371 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b/papers/coloring_nested_tire_graphs/paper.tex index db444fb..2f64217 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -87,8 +87,12 @@ with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ and $G$ has $2n - 4$ triangular faces. \begin{definition}[Level source] -A \emph{level source} of $G$ is any vertex $v \in V$; we write -$S = \{v\}$ for the level-0 source. +A \emph{level source} of $G$ is a set $S \subseteq V$ that is either +\begin{itemize} +\item a single vertex $\{v\}$ (a \emph{vertex source}), or +\item a set inducing a simple cycle in $G$ --- i.e.\ $G[S]$ is a + simple cycle of length $\geq 3$ (a \emph{cycle source}). +\end{itemize} \end{definition} \begin{definition}[Levels] @@ -932,6 +936,186 @@ of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty interior, $T_p$ has exactly one child. \end{remark} +\begin{theorem}[Tire-tree decomposition] +\label{thm:tire-tree-decomposition} +Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and let +$v_0 \in V(G)$. The tree of tire treads $\mathcal{T}(G, \{v_0\})$ of +Theorem~\ref{thm:tread-tree} \emph{decomposes $G$ into nested tires}: +it is a finite rooted tree, rooted at the depth-$0$ tread containing +$v_0$, whose nodes (tire treads) partition the bounded faces of $G$ +(Theorem~\ref{thm:tread-partition}). + +This decomposition is moreover \emph{self-similar}. For any tread $T$ +in $\mathcal{T}(G, \{v_0\})$ at depth $d \ge 1$, with outer-boundary +cycle $C_T := B_{\mathrm{out}}^{(T)}$, let $G_T$ be the sub-graph of $G$ +induced by $C_T$ together with all vertices of $G$ lying in the closed +planar region $R_T \subset |\Pi_G|$ bounded by $C_T$ on the side of +$C_T$ away from $v_0$. Then: +\begin{enumerate} +\item[(D1)] $G_T$, with the embedding inherited from $\Pi_G$, is a + \emph{triangulated disk}: every bounded face is a triangle, + and the outer face is bounded by $C_T$. +\item[(D2)] Taking $C_T$ as a cycle source of $G_T$ (so $C_T$ has + level $0$ in $G_T$ and the BFS-from-$C_T$ levels in $G_T$ + equal $\ell_G(\cdot) - d$ on $V(G_T)$), the construction of + Theorem~\ref{thm:tread-tree} extends to give a rooted tree + of tire treads $\mathcal{T}(G_T, C_T)$ whose depth-$0$ root + tread has $B_{\mathrm{out}} = C_T$ and inner outerplanar + graph $O = O^{(T)}$. +\item[(D3)] $\mathcal{T}(G_T, C_T)$ is canonically iso to the sub-tree + of $\mathcal{T}(G, \{v_0\})$ rooted at $T$, preserving + outer-boundary cycles, inner outerplanar graphs, and the + parent--child face correspondence. +\end{enumerate} + +In short: pick any vertex $v_0 \in V(G)$ to root the global tree +$\mathcal{T}(G, \{v_0\})$ describing the whole graph; pick any tread $T$ +in this tree; then $T$ is itself the root of a local tree +$\mathcal{T}(G_T, C_T)$ describing the triangulated disk of $G$ inside +$C_T$, with $C_T$ as cycle source. Maximal planar graphs decompose +into nested trees of tire treads. +\end{theorem} + +\begin{proof} +\emph{Decomposition.} Theorem~\ref{thm:tread-tree} gives the rooted +tree structure of $\mathcal{T}(G, \{v_0\})$, with root the depth-$0$ +tread containing $v_0$; Theorem~\ref{thm:tread-partition} gives that +its tire treads partition the bounded faces of $G$. Finiteness of +the tree is immediate from finiteness of $G$. + +\emph{(D1) $G_T$ is a triangulated disk.} By +Lemma~\ref{lem:tire-component} applied to the component of $G'_d$ that +gives rise to $T$, the outer boundary $C_T = B_{\mathrm{out}}^{(T)}$ is +a simple cycle in $L_d^G$. By the Jordan curve theorem, $C_T$ +separates $|\Pi_G| \setminus C_T$ into two open regions; $R_T$ is the +closure of the one not containing $v_0$. The bounded faces of $G_T$ in +its inherited embedding are exactly the bounded faces of $G$ contained +in $R_T$, each of which is a triangle since $G$ is a triangulation. +The unbounded face of $G_T$'s embedding is the complement of $R_T$, +whose boundary is $C_T$. + +\emph{(D2) Level shift.} We show $\mathrm{dist}_{G_T}(v, C_T) = +\ell_G(v) - d$ for every $v \in V(G_T)$. When $v \in C_T$ both sides +equal $0$, so fix $v \in V(G_T) \setminus C_T$. + +\smallskip + +\emph{Step 1: $\mathrm{dist}_G(v, C_T) = \ell_G(v) - d$.} A shortest +$G$-path from $v$ to $v_0$ must visit $C_T$, since $v$ and $v_0$ lie in +different open regions of $|\Pi_G| \setminus C_T$; let $w$ be its first +$C_T$-vertex. The $v$-to-$w$ sub-path has length $\ge \mathrm{dist}_G(v, +C_T)$ and the $w$-to-$v_0$ sub-path has length $\ell_G(w) = d$, so +$\ell_G(v) \ge \mathrm{dist}_G(v, C_T) + d$. Conversely, concatenating +a shortest $G$-path from $v$ to a nearest $C_T$-vertex $w'$ with a +shortest $G$-path from $w'$ to $v_0$ gives a $v$-to-$v_0$ path of +length $\mathrm{dist}_G(v, C_T) + d$, so $\ell_G(v) \le +\mathrm{dist}_G(v, C_T) + d$. + +\smallskip + +\emph{Step 2: $\mathrm{dist}_{G_T}(v, C_T) = \mathrm{dist}_G(v, C_T)$.} +The inequality $\ge$ is automatic since $G_T \subseteq G$. For $\le$, +pick a shortest $G$-path $\pi$ from $v$ to $C_T$; we may assume $\pi$ +has no internal vertex in $C_T$ (truncate otherwise). Any internal +vertex of $\pi$ then lies in the same open region of $|\Pi_G| \setminus +C_T$ as $v$, i.e.\ in $R_T \setminus C_T \subseteq V(G_T)$; every edge +of $\pi$ has both endpoints in $V(G_T)$ and so lies in $E(G_T)$. Hence +$\pi$ is a path in $G_T$ realising $\mathrm{dist}_G(v, C_T)$. + +\smallskip + +Combining the two steps yields $\mathrm{dist}_{G_T}(v, C_T) = \ell_G(v) +- d$, as claimed. + +\emph{(D3) Tree iso.} By (D2), $L_k^{G_T} = L_{d+k}^G \cap V(G_T)$ for +every $k \ge 0$. For a bounded face $f$ of $G_T$, dual depth in $G_T$ +equals $\min_{u \in V(f)} \ell_{G_T}(u) = \min_{u \in V(f)} \ell_G(u) - +d = \delta_G(d_f) - d$. Hence the inner-dual subgraph $(G_T)'_{k}$ at +depth $k$ in $G_T$ is the induced subgraph of $G'_{d+k}$ on the faces +of $G$ lying in $R_T$, and two such faces are dual-adjacent in $G_T'$ +iff they are dual-adjacent in $G'$ (the shared edge is in $E(G_T)$). + +\smallskip + +\emph{Step 3: components of $(G_T)'_{k}$ are precisely the depth-$(d+k)$ +descendants of $T$ in $\mathcal{T}(G, \{v_0\})$.} We show by induction +on $k$ that a component $C'$ of $G'_{d+k}$ has $F_{C'} \subseteq R_T$ +iff $C'$ is a depth-$(d+k)$ descendant of $T$. + +For $k = 0$: the components of $G'_d$ are the depth-$d$ treads; the +component giving rise to $T$ has its faces in $T$'s tread region +$R \subseteq R_T$, while any other depth-$d$ tread $T''$ has +$C_{T''}$ disjoint from $C_T$ and lying in a different bounded face of +$O^{(T_p'')}$ at depth $d - 1$, hence $R_{T''} \cap R_T = \emptyset$. + +For $k \ge 1$: by Theorem~\ref{thm:tread-tree}, each component $C'$ of +$G'_{d+k}$ has a unique parent $C'_p$ at depth $d+k-1$, with +$B_{\mathrm{out}}^{(C')}$ bounding a face of $O^{(C'_p)}$; equivalently +$R_{C'}$ lies inside that bounded face, hence inside $R_{C'_p}$. By +the induction hypothesis $R_{C'_p} \subseteq R_T$ iff $C'_p$ is a +descendant of $T$ at depth $d+k-1$, and $R_{C'} \subseteq R_{C'_p}$, so +$R_{C'} \subseteq R_T$ iff $C'$ is a descendant of $T$ at depth $d+k$. + +\smallskip + +\emph{Step 4: tread data and child--face correspondence.} The +Tire-component lemma (Lemma~\ref{lem:tire-component}) and the +source-side simple-cycle property +(Proposition~\ref{prop:no-level-d-pinch}) extend verbatim to the +cycle-sourced triangulated disk $(G_T, C_T)$: the proofs use only +the triangular structure of bounded faces, the local arrangement of +faces around each vertex's rotation, and the connectivity of the BFS +ball $G_T[L_{

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