From c2961b1833ddcf57308782a4ae6d14fa8c5b1ea9 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 17:04:13 -0400 Subject: [PATCH] coloring_nested_tire_graphs: prove no level-d pinches (Prop 1.7) Adds Proposition 1.7 (Source-side simple-cycle property): in any maximal planar graph G with single-vertex source v_0, for any vertex v at level d and any connected component C' of G'_d incident to v, the depth-d faces of F_{C'} at v form a single contiguous arc in v's rotation in Pi_G. Equivalently: the source-side boundary of R_{C'} is always a simple cycle in L_d, with no cut-vertices at level d. Proof: contradiction via Jordan curve theorem. If two arcs of depth-d faces at v exist, pick level-(d-1) neighbours p, q of v in the two gaps. The BFS ball G[L_{ p -> P -> q -> v is a simple cycle W (since v is the only vertex at level >= d on it). W separates the plane into two regions; the two arcs at v lie on opposite sides. Any dual path of depth-d faces from one arc to the other must avoid v on its intermediate faces, but consecutive intermediate faces share edges entirely in L_{>= d}, so the dual path stays on one side of W. This contradicts the endpoints being on opposite sides. Lemma 1.8 (the tire-component lemma) now cites Proposition 1.7 to justify that B_out is always a simple cycle. Level-(d+1) pinches (cut-vertices of O) remain allowed and are accommodated by the relaxed Definition 1.5. The empirical search (n in [7, 12], 47k + 276k = 323k components, 13k+ pinches all level-(d+1) cut-vertices of O, zero level-d pinches) is now subsumed by the structural proof. Co-Authored-By: Claude Opus 4.7 --- .../experiments/level_d_pinch_attempt.py | 93 ++++++++++++++++++ papers/coloring_nested_tire_graphs/paper.aux | 13 +-- papers/coloring_nested_tire_graphs/paper.log | 55 ++++++----- papers/coloring_nested_tire_graphs/paper.pdf | Bin 483570 -> 492830 bytes papers/coloring_nested_tire_graphs/paper.tex | 78 ++++++++++++++- 5 files changed, 202 insertions(+), 37 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/experiments/level_d_pinch_attempt.py diff --git a/papers/coloring_nested_tire_graphs/experiments/level_d_pinch_attempt.py b/papers/coloring_nested_tire_graphs/experiments/level_d_pinch_attempt.py new file mode 100644 index 0000000..475a634 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/level_d_pinch_attempt.py @@ -0,0 +1,93 @@ +"""Attempt to construct a level-d pinch in a maximal planar graph. + +Construction: + v_0 at center, pentagon a_1, ..., a_5 around v_0. + Outside the pentagon, vertex v adjacent to a_1, a_3 (non-adjacent + pentagon vertices). The "lobe" between v and the pentagon edge + a_1-a_2-a_3 contains a_2 and is triangulated by adding vertex x + in the lobe, x adjacent to a_1, a_2, a_3, v. On the OTHER side of + v (outside the V formed by edges v-a_1, v-a_3), we triangulate by + adding w_1 adjacent to a_1, a_3, a_4, a_5, v. + +Vertex labels: + 0 = v_0 (source) + 1..5 = a_1..a_5 (pentagon, level 1) + 6 = v (level 2) + 7 = x (level 2) + 8 = w_1 (level 2) + +If this is planar and is a maximal planar graph, then v has level-1 +neighbours a_1=1, a_3=3 in DIFFERENT arcs of v's rotation, separated +by level-2 neighbours x=7 on the lobe side and w_1=8 on the outer side. +This would be a level-d (d=2) pinch. +""" +import os +import sys +from collections import defaultdict +from sage.all import Graph + +EDGES = [ + (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), + (1, 2), (2, 3), (3, 4), (4, 5), (5, 1), + (6, 1), (6, 3), (6, 7), (6, 8), + (7, 1), (7, 2), (7, 3), + (8, 1), (8, 3), (8, 5), (8, 4), +] + + +def main(): + G = Graph(EDGES) + print(f"n = {G.order()}, m = {G.size()}") + print(f"Max planar would need m = 3n - 6 = {3 * G.order() - 6}") + print(f"Planar: {G.is_planar()}") + if not G.is_planar(): + print("Not planar -- construction failed.") + return + + G.is_planar(set_embedding=True) + emb = G.get_embedding() + print(f"Rotation around v=6: {emb[6]}") + + dist = G.shortest_path_lengths(0) + print(f"BFS distances from source v_0=0: {dict(dist)}") + print() + + # Pretty-print rotation of vertex 6 with levels + rot_6 = emb[6] + print(f"v=6's rotation in Pi_G: {rot_6}") + print(f" levels: {[dist[u] for u in rot_6]}") + + # Check faces around vertex 6 + faces = G.faces(embedding=emb) + faces_at_6 = [] + for f in faces: + verts = [e[0] for e in f] + if 6 in verts: + d = min(dist[v] for v in verts) + faces_at_6.append((verts, d)) + print(f"\nFaces at vertex 6:") + for f, d in faces_at_6: + print(f" {f} depth={d}") + + # Check the depth-(d-1=1)? No, we want vertex 6 at level 2. + # We're checking: vertex 6 (level 2) with level-1 neighbors {1, 3}. + # Are they in the same arc of vertex 6's rotation? + level1_nbrs = [u for u in rot_6 if dist[u] == 1] + print(f"\nLevel-1 neighbours of v=6: {level1_nbrs}") + if len(level1_nbrs) > 1: + # Find arcs + positions = [i for i, u in enumerate(rot_6) if dist[u] == 1] + print(f" positions in rotation: {positions}") + # Are they contiguous? Count runs + in_l1 = [dist[u] == 1 for u in rot_6] + n_runs = sum(1 for i in range(len(in_l1)) + if in_l1[i] and not in_l1[(i - 1) % len(in_l1)]) + print(f" number of arcs of level-1 neighbours: {n_runs}") + if n_runs > 1: + print(" ==> LEVEL-d PINCH! Counterexample constructed.") + else: + print(" ==> single contiguous arc. No pinch.") + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 86a1424..7f0b0f7 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -4,19 +4,20 @@ \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:tire-graph}{{1.5}{2}} -\citation{bauerfeld-pds} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{3}} \newlabel{rem:tire-counts}{{1.6}{3}} -\newlabel{lem:tire-component}{{1.7}{3}} +\newlabel{prop:no-level-d-pinch}{{1.7}{3}} \citation{bauerfeld-pds} +\newlabel{lem:tire-component}{{1.8}{4}} +\citation{bauerfeld-pds} +\newlabel{rem:tire-component-degenerate}{{1.9}{5}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tire-component-degenerate}{{1.8}{5}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.9}{5}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } -\gdef \@abspage@last{5} +\newlabel{rem:tire-no-extra-hypotheses}{{1.10}{6}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } +\gdef \@abspage@last{6} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 4e088e5..699a6c6 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 16:37 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 17:03 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -207,36 +207,39 @@ File: fig_tire_example.png Graphic file (type png) Package pdftex.def Info: fig_tire_example.png used on input line 160. 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