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_{<d}] is connected so admits a simple
path P from p to q.  The closed walk v -> 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 <noreply@anthropic.com>

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()

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}{{1.10}{6}}
-\newlabel{rem:tire-no-extra-hypotheses}{{1.9}{5}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
+\gdef \@abspage@last{6}
-\gdef \@abspage@last{5}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -178,6 +178,68 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
 triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
 \end{remark}
 
+\begin{proposition}[Source-side simple-cycle property]
+\label{prop:no-level-d-pinch}
+Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
+single-vertex source $v_0$.  Let $d \geq 1$, $v \in L_d$, and let
+$C'$ be a connected component of $G'_d$ such that $v$ is incident to
+some face in $F_{C'}$.  Then the depth-$d$ faces in $F_{C'}$ incident
+to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$.
+
+Equivalently: for any such component, the source-side boundary of
+$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$).
+\end{proposition}
+
+\begin{proof}
+Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$
+lie in two or more disjoint arcs of $v$'s rotation.  Adjacent vertices
+in $G$ differ in level by at most $1$, so a face at $v$ has depth
+exactly $d$ iff both other vertices have level $\geq d$, and depth
+$\leq d-1$ iff at least one has level $d-1$.  Hence the gaps between
+the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of
+$v$, occurring in at least two disjoint arcs of $v$'s rotation.  Pick
+$p$ in one such gap and $q$ in another.
+
+The BFS ball $G[L_{<d}]$ is connected, so there exists a simple path
+$P$ in $G[L_{<d}]$ from $p$ to $q$.  Define the closed walk
+\[
+    W \;:=\; v \to p \to P \to q \to v.
+\]
+Every vertex of $P$ lies in $L_{<d}$, while $\ell(v) = d$, so $v$ is
+distinct from every vertex of $P$; $P$ is simple, so its internal
+vertices are distinct; and $p \neq q$ since they lie in different gaps.
+Hence $W$ is a simple cycle in $G$.
+
+By the Jordan curve theorem, the planar embedding of $W$ divides $\Pi_G$
+into two regions.  In $v$'s rotation, the edges $v-p$ and $v-q$ lie at
+two specific positions, and they split the rotation into two arcs;
+each arc lies in one of the two regions determined by $W$.  By choice
+of $p, q$, the two arcs of depth-$d$ faces at $v$ in $F_{C'}$ lie in
+different regions of $W$ (i.e., one arc on each side).
+
+Since $C'$ is connected in $G'$ and contains depth-$d$ faces in both
+arcs, there is a dual path $f_1, f_2, \ldots, f_k$ in $G'_d$ with
+$f_1, f_k \in F_{C'}$ incident to $v$ in different arcs, and with the
+intermediate faces $f_2, \ldots, f_{k-1}$ not incident to $v$ (a
+shortest such dual path).  Consecutive faces $f_i, f_{i+1}$ share an
+edge $e_i$ of $G$; for $i \geq 2$, both endpoints of $e_i$ lie in
+$L_{\geq d}$ (since neither $f_i$ nor $f_{i+1}$ is incident to $v$,
+all six vertices of these two triangles lie in $L_{\geq d}$).  In
+particular, $e_i$ shares no endpoint with $W$ except possibly $v$ ---
+and $v$ is excluded from $f_2, \ldots, f_{k-1}$.
+
+A planar edge with neither endpoint on a simple closed planar curve
+$W$ has both of its incident faces on the same side of $W$.  Applying
+this to each $e_i$ ($i \geq 2$) inductively: starting from $f_2$ on
+the same side of $W$ as $f_1$ (their shared edge $e_1 = w-w'$ opposite
+to $v$ in $f_1$ has $w, w' \in L_{\geq d}$ and hence is not on $W$),
+the path $f_2 \to f_3 \to \cdots \to f_{k-1} \to f_k$ stays on one
+side of $W$.
+
+But $f_1$ and $f_k$ lie on different sides of $W$ (by construction),
+contradicting the conclusion that the entire path lies on one side.
+\end{proof}
+
 \begin{lemma}[Tire-component lemma]
 \label{lem:tire-component}
 Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level
@@ -249,11 +311,17 @@ is monochromatic in level.
 \emph{Boundary structure.}  Each connected component of
 $\partial R_{C'}$ traces a closed walk in $G$ that, by the
 monochromaticity above, lies entirely in $L_d$ or entirely in
-$L_{d+1}$.  At a vertex $v \in V_{C'}$ where the faces of $F_{C'}$
+$L_{d+1}$.  By Proposition~\ref{prop:no-level-d-pinch}, the depth-$d$
-split into multiple arcs of $v$'s rotation, the boundary walk visits
+faces of $F_{C'}$ at any $v \in L_d \cap V_{C'}$ form a single
-$v$ correspondingly many times; this occurs precisely when $v$ is a
+contiguous arc in $v$'s rotation, so the source-side boundary walk
-cut-vertex of the induced subgraph on its level (either $L_d$ or
+visits each $L_d$-vertex of $V_{C'}$ exactly once: it is a simple
-$L_{d+1}$).
+cycle.  At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
+may split into multiple arcs of $v$'s rotation; this corresponds
+exactly to $v$ being a cut-vertex of $O$, and the inner-side
+boundary walk visits $v$ correspondingly many times --- which is
+already accommodated by Definition~\ref{def:tire-graph} (where
+$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
+necessarily a simple cycle).
 
 \emph{Outer boundary.}  Because $S$ lies on the outer face of $\Pi_G$,
 the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$