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 0000000..a80da8e Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_seam_construction.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index ab4e642..4a9e157 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -31,8 +31,12 @@ \newlabel{thm:tread-tree}{{1.17}{10}} \newlabel{rem:tree-multiple-children}{{1.18}{11}} \newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} -\newlabel{conj:universal-nesting}{{1.20}{12}} -\newlabel{rem:nesting-motivation}{{1.21}{12}} +\newlabel{def:tree-iso-O-preserved}{{1.20}{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{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -41,5 +45,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{13}{}\protected@file@percent } -\gdef \@abspage@last{13} +\newlabel{rem:nesting-motivation}{{1.24}{14}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent } +\gdef \@abspage@last{14} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 64faef6..b7995d2 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) 27 MAY 2026 03:32 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 04:24 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -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'. -[7] [8] [9] [10] [11] [12] [13] (./paper.aux) ) +[7] [8] [9] [10] [11] +Overfull \hbox (2.78796pt too wide) in paragraph at lines 1013--1017 +[]\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] + +File: fig_seam_construction.png Graphic file (type png) + +Package pdftex.def Info: fig_seam_construction.png used on input line 1038. +(pdftex.def) Requested size: 341.9989pt x 185.89983pt. + [13 <./fig_seam_construction.png>] +Overfull \hbox (2.06076pt too wide) in paragraph at lines 1106--1120 +[]\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) ) Here is how much of TeX's memory you used: - 14050 strings out of 478268 - 279277 string characters out of 5846347 - 563864 words of memory out of 5000000 - 31874 multiletter control sequences out of 15000+600000 + 14061 strings out of 478268 + 279583 string characters out of 5846347 + 563909 words of memory out of 5000000 + 31884 multiletter control sequences out of 15000+600000 478218 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 84i,12n,89p,1156b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (13 pages, 623232 bytes). + + +Output written on paper.pdf (14 pages, 927914 bytes). PDF statistics: - 181 PDF objects out of 1000 (max. 8388607) - 110 compressed objects within 2 object streams + 186 PDF objects out of 1000 (max. 8388607) + 112 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) - 23 words of extra memory for PDF output out of 10000 (max. 10000000) + 28 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 0c7a22c..690d423 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 81e9dac..9d72f79 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -950,45 +950,133 @@ This is the structural setup underlying the chain-pigeonhole program for tire treads. \end{remark} -\begin{conjecture}[Universal nesting of tire-tread trees, sketch] -\label{conj:universal-nesting} -For any two rooted trees of tire treads -$\mathcal{T}_1 = \mathcal{T}(G_1, S_1)$ and -$\mathcal{T}_2 = \mathcal{T}(G_2, S_2)$ arising from maximal planar -graphs $G_1, G_2$ with respective single-vertex level sources -$S_1, S_2$, the following holds: $\mathcal{T}_1$ \emph{nests} -into $\mathcal{T}_2$. - -By ``$\mathcal{T}_1$ nests into $\mathcal{T}_2$'' we mean: +\begin{definition}[Iso of rooted trees of tire treads; combinatorial, $O$-preserved] +\label{def:tree-iso-O-preserved} +Let $\mathcal{T}_1, \mathcal{T}_2$ be rooted trees of tire treads. A +\emph{combinatorial, $O$-preserved iso} from $\mathcal{T}_1$ to +$\mathcal{T}_2$ is a pair $(\varphi, \{\varphi_T\}_{T \in \mathcal{T}_1})$ +satisfying: \begin{itemize} -\item Choose any tire tread $T \in \mathcal{T}_2$ and any non-trivial - bounded face $f$ of its inner outerplanar graph $O^{(T)}$ - (i.e.\ a face whose interior currently contains depth-$\ge d+2$ - vertices of $G_2$, where $d = \mathrm{depth}(T)$). -\item Then there exists a maximal planar graph $\tilde G$ with - level source $\tilde S$ such that: - \begin{enumerate} - \item[(N1)] $\mathcal{T}(\tilde G, \tilde S)$ contains - $\mathcal{T}_2$ as a sub-tree (with every - tire tread of $\mathcal{T}_2$ preserved - combinatorially and embedded); - \item[(N2)] the sub-tree of $\mathcal{T}(\tilde G, \tilde S)$ - rooted at the child of $T$ corresponding to face - $f$ is isomorphic, as a rooted tree of tire treads, - to $\mathcal{T}_1$. - \end{enumerate} +\item $\varphi : \mathcal{T}_1 \to \mathcal{T}_2$ is a rooted-tree iso + (root to root, parent edges to parent edges); +\item for each tread $T \in \mathcal{T}_1$, $\varphi_T : O^{(T)} \to + O^{(\varphi(T))}$ is an iso of plane outerplanar graphs --- in + particular, the set of bounded faces of $O^{(T)}$ is sent + bijectively to that of $O^{(\varphi(T))}$, with cyclic structure + of each face preserved; +\item the child--face correspondence commutes with $\varphi$: if $T_c$ + is the child of $T$ at the bounded face $f$ of $O^{(T)}$, then + $\varphi(T_c)$ is the child of $\varphi(T)$ at the bounded face + $\varphi_T(f)$ of $O^{(\varphi(T))}$. +\end{itemize} +The outer boundaries $B_{\mathrm{out}}^{(T)}$ are \emph{not} required to +correspond. In particular, the root tread's outer boundary may be +degenerate (a single vertex) in $\mathcal{T}_1$ and a simple cycle in +$\mathcal{T}_2$; 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} -\medskip - -Informally: any tree of tire treads can be ``inserted'' into any -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 +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 +of $\mathcal{T}_1$. The class of trees of tire treads is \emph{closed under composition} by face-slot insertion. \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} The conjectured closure under nesting carries two structural 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 of $G_1$ and $G_2$ alone (via the parent--child consistency constraints of Remark~\ref{rem:tree-coloring-factorisation}), - then $4$-colourability propagates through nesting. A - minimum $4$CT counterexample (if it exists) would have to be + then $4$-colourability propagates through nesting. A minimum + $4$CT counterexample (if it exists) would have to be \emph{irreducible} under such nesting --- it could not be decomposed into strictly smaller trees of tire treads whose colourings combine to a colouring of the whole. @@ -1011,12 +1099,44 @@ implications for the Four Colour Theorem programme: composition rule. \end{itemize} -Open questions include: which precise notion of ``isomorphic as -rooted trees of tire treads'' should be used (combinatorial, -geometric, or up to embedding)? Does the nested triangulation -$\tilde G$ admit a constructive description from $G_1, G_2$ and -the choice of face $f$? And does nesting respect Birkhoff's -internally $6$-connected condition for minimum counterexamples? +\medskip + +Open questions: +\begin{itemize} +\item \emph{Candidate seam construction.} A natural candidate for + $H_k$ 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}