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coloring_nested_tire_graphs: pin nesting iso, factor seam lemma, add figure

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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 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 04:30:48 -04:00
parent 6413560a7b
commit 388ab19db9
6 changed files with 420 additions and 81 deletions
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papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py
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"""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}')
papers/coloring_nested_tire_graphs/fig_seam_construction.png
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papers/coloring_nested_tire_graphs/paper.aux
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@@ -31,8 +31,12 @@
\newlabel{thm:tread-tree}{{1.17}{10}} \newlabel{thm:tread-tree}{{1.17}{10}}
\newlabel{rem:tree-multiple-children}{{1.18}{11}} \newlabel{rem:tree-multiple-children}{{1.18}{11}}
\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} \newlabel{rem:tree-coloring-factorisation}{{1.19}{12}}
\newlabel{conj:universal-nesting}{{1.20}{12}} \newlabel{def:tree-iso-O-preserved}{{1.20}{12}}
\newlabel{rem:nesting-motivation}{{1.21}{12}} \newlabel{conj:universal-nesting}{{1.21}{12}}
\newlabel{conj:seam-realizability}{{1.22}{12}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Seam realizability for a small example. $(a)$ A stacked-ring triangulation $G_1$ with single-vertex source $S_1 = \{0\}$ and concentric levels $L_1, L_2, L_3$; its tree of tire treads is the chain $T_0 \to T_1 \to T_2$ with $O^{(T_d)} = G_1[L_{d+1}]$ a $3$-cycle on each tread. $(b)$ The apex-removal seam construction $H_5 = (G_1 \setminus S_1) \cup A_5$, re-embedded so that the former fan-face around $S_1$ becomes the outer face (with $L_1$ now the outermost $G_1$-derived ring and $L_3$ innermost), and with an annular triangulation $A_5$ (orange) attaching to a fresh $5$-cycle $\partial H_5$ (red). Vertex labels show $\mathrm {BFS}_{\partial H_5}$ levels in $H_5$: they agree with $\ell _{G_1}$ on $V(G_1) \setminus \{S_1\}$, so $\mathcal {T}(H_5, \partial H_5)$ is iso (combinatorial, $O$-preserved) to $\mathcal {T}(G_1, S_1)$.}}{13}{}\protected@file@percent }
\newlabel{fig:seam-construction}{{5}{13}}
\newlabel{rem:seam-reduces-nesting}{{1.23}{13}}
\bibcite{tait-original}{1} \bibcite{tait-original}{1}
\bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3} \bibcite{bauerfeld-nested-tire-duals}{3}
@@ -41,5 +45,6 @@
\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{13}{}\protected@file@percent } \newlabel{rem:nesting-motivation}{{1.24}{14}}
\gdef \@abspage@last{13} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
\gdef \@abspage@last{14}
papers/coloring_nested_tire_graphs/paper.log
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entering extended mode entering extended mode
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@@ -511,45 +511,64 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
[7] [8] [9] [10] [11] [12] [13] (./paper.aux) ) [7] [8] [9] [10] [11]
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[]\OT1/cmr/bx/n/10 Conjecture 1.22 \OT1/cmr/m/n/10 (Seam re-al-iz-abil-ity; tec
h-ni-cal core of nest-ing)\OT1/cmr/bx/n/10 . []\OT1/cmr/m/it/10 Let $\OMS/cmsy/
m/n/10 T[] \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 T\OT1/cmr/m/n/10 (\OML/cmm/m/it/1
0 G[]; S[]\OT1/cmr/m/n/10 )$
[]
[12]
<fig_seam_construction.png, id=72, 1124.2pt x 611.083pt>
File: fig_seam_construction.png Graphic file (type png)
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Package pdftex.def Info: fig_seam_construction.png used on input line 1038.
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[13 <./fig_seam_construction.png>]
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[]\OT1/cmr/m/it/10 Candidate seam con-struc-tion. \OT1/cmr/m/n/10 A nat-u-ral c
an-di-date for $\OML/cmm/m/it/10 H[]$ \OT1/cmr/m/n/10 in Con-jec-ture 1.22[]
[]
[14] (./paper.aux) )
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Output written on paper.pdf (13 pages, 623232 bytes).
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -950,45 +950,133 @@ This is the structural setup underlying the chain-pigeonhole
program for tire treads. program for tire treads.
\end{remark} \end{remark}
\begin{conjecture}[Universal nesting of tire-tread trees, sketch] \begin{definition}[Iso of rooted trees of tire treads; combinatorial, $O$-preserved]
\label{conj:universal-nesting} \label{def:tree-iso-O-preserved}
For any two rooted trees of tire treads Let $\mathcal{T}_1, \mathcal{T}_2$ be rooted trees of tire treads. A
$\mathcal{T}_1 = \mathcal{T}(G_1, S_1)$ and \emph{combinatorial, $O$-preserved iso} from $\mathcal{T}_1$ to
$\mathcal{T}_2 = \mathcal{T}(G_2, S_2)$ arising from maximal planar $\mathcal{T}_2$ is a pair $(\varphi, \{\varphi_T\}_{T \in \mathcal{T}_1})$
graphs $G_1, G_2$ with respective single-vertex level sources satisfying:
$S_1, S_2$, the following holds: $\mathcal{T}_1$ \emph{nests}
into $\mathcal{T}_2$.
By ``$\mathcal{T}_1$ nests into $\mathcal{T}_2$'' we mean:
\begin{itemize} \begin{itemize}
\item Choose any tire tread $T \in \mathcal{T}_2$ and any non-trivial \item $\varphi : \mathcal{T}_1 \to \mathcal{T}_2$ is a rooted-tree iso
bounded face $f$ of its inner outerplanar graph $O^{(T)}$ (root to root, parent edges to parent edges);
(i.e.\ a face whose interior currently contains depth-$\ge d+2$ \item for each tread $T \in \mathcal{T}_1$, $\varphi_T : O^{(T)} \to
vertices of $G_2$, where $d = \mathrm{depth}(T)$). O^{(\varphi(T))}$ is an iso of plane outerplanar graphs --- in
\item Then there exists a maximal planar graph $\tilde G$ with particular, the set of bounded faces of $O^{(T)}$ is sent
level source $\tilde S$ such that: bijectively to that of $O^{(\varphi(T))}$, with cyclic structure
\begin{enumerate} of each face preserved;
\item[(N1)] $\mathcal{T}(\tilde G, \tilde S)$ contains \item the child--face correspondence commutes with $\varphi$: if $T_c$
$\mathcal{T}_2$ as a sub-tree (with every is the child of $T$ at the bounded face $f$ of $O^{(T)}$, then
tire tread of $\mathcal{T}_2$ preserved $\varphi(T_c)$ is the child of $\varphi(T)$ at the bounded face
combinatorially and embedded); $\varphi_T(f)$ of $O^{(\varphi(T))}$.
\item[(N2)] the sub-tree of $\mathcal{T}(\tilde G, \tilde S)$ \end{itemize}
rooted at the child of $T$ corresponding to face The outer boundaries $B_{\mathrm{out}}^{(T)}$ are \emph{not} required to
$f$ is isomorphic, as a rooted tree of tire treads, correspond. In particular, the root tread's outer boundary may be
to $\mathcal{T}_1$. degenerate (a single vertex) in $\mathcal{T}_1$ and a simple cycle in
\end{enumerate} $\mathcal{T}_2$; this is essential because the root tread of a
\emph{sub-tree} of $\mathcal{T}(\tilde G, \tilde S)$ inherits a
non-degenerate $B_{\mathrm{out}}$ from its parent, even when it is iso
to a tree arising from a single-vertex level source.
\end{definition}
\begin{conjecture}[Universal nesting of tire-tread trees]
\label{conj:universal-nesting}
Let $\mathcal{T}_2 = \mathcal{T}(G_2, S_2)$ be a tree of tire treads
arising from a maximal planar $G_2$ with single-vertex level source
$S_2$. Let $T \in \mathcal{T}_2$ be a tread at depth $d$, and let $f$
be a non-trivial bounded face of $O^{(T)}$ (i.e.\ a face whose interior
contains depth-$\ge d+2$ vertices of $G_2$). Let $\mathcal{T}_1 =
\mathcal{T}(G_1, S_1)$ be any other tree of tire treads.
Then there exists a maximal planar graph $\tilde G$ with single-vertex
level source $\tilde S$ such that:
\begin{itemize}
\item[(N1)] $\mathcal{T}(\tilde G, \tilde S)$ contains, as a rooted
sub-tree, an iso copy (in the sense of
Definition~\ref{def:tree-iso-O-preserved}) of the
truncation $\mathcal{T}_2 \setminus \mathrm{Desc}(T, f)$
obtained from $\mathcal{T}_2$ by deleting the descendant
sub-tree of $T$ at face $f$;
\item[(N2)] the sub-tree of $\mathcal{T}(\tilde G, \tilde S)$ rooted at
the (new) child of $T$'s image at (the image of) $f$ is iso,
in the sense of Definition~\ref{def:tree-iso-O-preserved},
to $\mathcal{T}_1$.
\end{itemize} \end{itemize}
\medskip Informally: trees of tire treads are closed under face-slot insertion,
where the slot at face $f$ in $\mathcal{T}_2$ is filled by the entirety
Informally: any tree of tire treads can be ``inserted'' into any of $\mathcal{T}_1$. The class of trees of tire treads is
non-trivial face slot of any other tree of tire treads, producing
a larger maximal planar graph whose tree of tire treads is the
nested combination. The class of trees of tire treads is
\emph{closed under composition} by face-slot insertion. \emph{closed under composition} by face-slot insertion.
\end{conjecture} \end{conjecture}
\begin{remark} \begin{conjecture}[Seam realizability; technical core of nesting]
\label{conj:seam-realizability}
Let $\mathcal{T}_1 = \mathcal{T}(G_1, S_1)$ be a tree of tire treads.
For every integer $k \ge 3$ there exists a planar graph $H_k$, embedded
in a closed disk $D \subset \mathbb{R}^2$ with $\partial D$ a $k$-cycle,
such that:
\begin{itemize}
\item[(S1)] $\partial H_k = \partial D$, as a cyclic sequence of $k$
vertices;
\item[(S2)] every bounded face of $H_k$ is a triangle;
\item[(S3)] BFS in $H_k$ from the cycle $\partial H_k$ assigns levels to
$V(H_k) \setminus V(\partial H_k)$, and the resulting rooted
tree of tire treads --- with the depth-$0$ tread taking
outer boundary $\partial H_k$ in place of a single-vertex
source --- is iso, in the sense of
Definition~\ref{def:tree-iso-O-preserved}, to
$\mathcal{T}_1$.
\end{itemize}
The construction $\mathcal{T}(H_k, \partial H_k)$ is the natural
extension of Theorem~\ref{thm:tread-tree} from single-vertex sources to
cycle sources: the depth-$0$ tread has non-degenerate
$B_{\mathrm{out}} = \partial H_k$ and the rest of the construction is
unchanged.
\end{conjecture}
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{fig_seam_construction.png}
\caption{Seam realizability for a small example. $(a)$ A stacked-ring
triangulation $G_1$ with single-vertex source $S_1 = \{0\}$ and concentric
levels $L_1, L_2, L_3$; its tree of tire treads is the chain $T_0 \to T_1
\to T_2$ with $O^{(T_d)} = G_1[L_{d+1}]$ a $3$-cycle on each tread.
$(b)$ The apex-removal seam construction
$H_5 = (G_1 \setminus S_1) \cup A_5$, re-embedded so that the former
fan-face around $S_1$ becomes the outer face (with $L_1$ now the
outermost $G_1$-derived ring and $L_3$ innermost), and with an annular
triangulation $A_5$ (orange) attaching to a fresh $5$-cycle $\partial H_5$
(red). Vertex labels show $\mathrm{BFS}_{\partial H_5}$ levels in $H_5$:
they agree with $\ell_{G_1}$ on $V(G_1) \setminus \{S_1\}$, so
$\mathcal{T}(H_5, \partial H_5)$ is iso (combinatorial, $O$-preserved) to
$\mathcal{T}(G_1, S_1)$.}
\label{fig:seam-construction}
\end{figure}
\begin{remark}[Nesting reduces to seam realizability]
\label{rem:seam-reduces-nesting}
Conjecture~\ref{conj:universal-nesting} follows from
Conjecture~\ref{conj:seam-realizability} by a direct gluing argument
within the framework of this paper. Briefly: given a disk realization
$H_k$ of $\mathcal{T}_1$ with $k = |C_f|$, where $C_f$ is the cycle
bounding $f$ in $O^{(T)}$, excise from $G_2$ all vertices and edges
strictly inside $f$, then glue $H_k$ into the resulting hole by
identifying $\partial H_k$ with $C_f$. The verification that the glued
graph $\tilde G$ is maximal planar, retains $\tilde S = S_2$ as a
single-vertex level source, and realizes the claimed nesting --- the
levels of $\tilde G$ from $S_2$ inside $f$ being just BFS-from-$C_f$ in
$H_k$ shifted by $d + 1$ --- is mechanical from
Theorems~\ref{thm:tread-partition},
\ref{thm:inner-dual-outerplanar},
\ref{thm:tread-tree} and the parent--child interface description of
Remark~\ref{rem:tree-coloring-factorisation}.
The substantive content of universal nesting thus sits entirely in
Conjecture~\ref{conj:seam-realizability}: given an arbitrary tree of
tire treads, can it be realized as the BFS-from-boundary tree of treads
of a triangulated planar disk, for every boundary length $k \ge 3$?
\end{remark}
\begin{remark}[Motivation and open questions]
\label{rem:nesting-motivation} \label{rem:nesting-motivation}
The conjectured closure under nesting carries two structural The conjectured closure under nesting carries two structural
implications for the Four Colour Theorem programme: implications for the Four Colour Theorem programme:
@@ -997,8 +1085,8 @@ implications for the Four Colour Theorem programme:
$\tilde G$ in (N1)--(N2) can be decided from the colourability $\tilde G$ in (N1)--(N2) can be decided from the colourability
of $G_1$ and $G_2$ alone (via the parent--child consistency of $G_1$ and $G_2$ alone (via the parent--child consistency
constraints of Remark~\ref{rem:tree-coloring-factorisation}), constraints of Remark~\ref{rem:tree-coloring-factorisation}),
then $4$-colourability propagates through nesting. A then $4$-colourability propagates through nesting. A minimum
minimum $4$CT counterexample (if it exists) would have to be $4$CT counterexample (if it exists) would have to be
\emph{irreducible} under such nesting --- it could not be \emph{irreducible} under such nesting --- it could not be
decomposed into strictly smaller trees of tire treads whose decomposed into strictly smaller trees of tire treads whose
colourings combine to a colouring of the whole. colourings combine to a colouring of the whole.
@@ -1011,12 +1099,44 @@ implications for the Four Colour Theorem programme:
composition rule. composition rule.
\end{itemize} \end{itemize}
Open questions include: which precise notion of ``isomorphic as \medskip
rooted trees of tire treads'' should be used (combinatorial,
geometric, or up to embedding)? Does the nested triangulation Open questions:
$\tilde G$ admit a constructive description from $G_1, G_2$ and \begin{itemize}
the choice of face $f$? And does nesting respect Birkhoff's \item \emph{Candidate seam construction.} A natural candidate for
internally $6$-connected condition for minimum counterexamples? $H_k$ in Conjecture~\ref{conj:seam-realizability} is the
\emph{apex-removal} construction:
$H_k = (G_1 \setminus S_1) \cup A_k$, where $A_k$ is a
triangulated annulus from the cycle $L_1^{(G_1)}$ to a fresh
$k$-cycle that serves as $\partial H_k$; the embedding is chosen
so the former fan-face around $S_1$ in $G_1$ becomes the outer
face. Showing that $\mathcal{T}(H_k, \partial H_k)$ is iso
(combinatorial, $O$-preserved) to $\mathcal{T}_1$ amounts to
verifying that BFS distances from $\partial H_k$ in $H_k$
reproduce $\ell_{G_1}(\cdot)$ on $V(G_1) \setminus \{S_1\}$ ---
which follows from the observation that every shortest path in
$H_k$ from a non-boundary vertex to $\partial H_k$ passes through
$L_1^{(G_1)}$.
\item \emph{$6$-connectivity preservation.} Does nesting respect
Birkhoff's internally $6$-connected condition for minimum $4$CT
counterexamples? The gluing seam $C_f \sim \partial H_k$ is
exactly the low-connectivity site, so even when $G_1, G_2$ are
internally $6$-connected the resulting $\tilde G$ is generically
not, absent further hypotheses on $(G_1, G_2, f, k)$. Identifying
sufficient conditions for $6$-connected-preserving nesting is the
relevant subproblem for the $4$CT application.
\item \emph{Stronger iso notions.}
Definition~\ref{def:tree-iso-O-preserved} allows
$B_{\mathrm{out}}^{(T)}$'s to differ. A strictly stronger
version of Conjecture~\ref{conj:universal-nesting} would require
$B_{\mathrm{out}}$'s to correspond as cycles, but this is
generically false: the cycle $C_f$ has fixed length $k$
determined by $G_2$, while the depth-$1$ cycle $L_1^{(G_1)}$ has
length $\deg_{G_1}(S_1)$ determined by $G_1$. The combinatorial,
$O$-preserved version of Definition~\ref{def:tree-iso-O-preserved}
is exactly the notion that allows the seam to absorb this length
mismatch.
\end{itemize}
\end{remark} \end{remark}