coloring_nested_tire_graphs: drop the universal-nesting block
Delete Definition 1.20 (iso of trees of tire treads), Conjecture 1.21 (universal nesting), Conjecture 1.22 (seam realizability), the seam-construction figure inclusion, Remark 1.23 (nesting reduces to seam), and Remark 1.24 (motivation / open questions). The paper now ends after Remark 1.19 (tree-coloring-factorisation). The fig_seam_construction.png file and its generator script remain in the repo as assets; nothing in the paper currently references them. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -30,13 +30,6 @@
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\newlabel{rem:count-general-outerplanar}{{1.16}{10}}
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\newlabel{thm:tread-tree}{{1.17}{10}}
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\newlabel{rem:tree-multiple-children}{{1.18}{11}}
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\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}}
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\newlabel{def:tree-iso-O-preserved}{{1.20}{12}}
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\newlabel{conj:universal-nesting}{{1.21}{12}}
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\newlabel{conj:seam-realizability}{{1.22}{12}}
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\@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 }
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\newlabel{fig:seam-construction}{{5}{13}}
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\newlabel{rem:seam-reduces-nesting}{{1.23}{13}}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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@@ -45,6 +38,6 @@
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{rem:nesting-motivation}{{1.24}{14}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
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\gdef \@abspage@last{14}
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\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent }
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[]\OT1/cmr/bx/n/10 Conjecture 1.22 \OT1/cmr/m/n/10 (Seam re-al-iz-abil-ity; tec
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h-ni-cal core of nest-ing)\OT1/cmr/bx/n/10 . []\OT1/cmr/m/it/10 Let $\OMS/cmsy/
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[]\OT1/cmr/m/it/10 Candidate seam con-struc-tion. \OT1/cmr/m/n/10 A nat-u-ral c
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an-di-date for $\OML/cmm/m/it/10 H[]$ \OT1/cmr/m/n/10 in Con-jec-ture 1.22[]
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@@ -950,196 +950,6 @@ This is the structural setup underlying the chain-pigeonhole
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program for tire treads.
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\end{remark}
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\begin{definition}[Iso of rooted trees of tire treads; combinatorial, $O$-preserved]
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\label{def:tree-iso-O-preserved}
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Let $\mathcal{T}_1, \mathcal{T}_2$ be rooted trees of tire treads. A
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\emph{combinatorial, $O$-preserved iso} from $\mathcal{T}_1$ to
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$\mathcal{T}_2$ is a pair $(\varphi, \{\varphi_T\}_{T \in \mathcal{T}_1})$
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satisfying:
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\begin{itemize}
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\item $\varphi : \mathcal{T}_1 \to \mathcal{T}_2$ is a rooted-tree iso
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(root to root, parent edges to parent edges);
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\item for each tread $T \in \mathcal{T}_1$, $\varphi_T : O^{(T)} \to
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O^{(\varphi(T))}$ is an iso of plane outerplanar graphs --- in
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particular, the set of bounded faces of $O^{(T)}$ is sent
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bijectively to that of $O^{(\varphi(T))}$, with cyclic structure
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of each face preserved;
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\item the child--face correspondence commutes with $\varphi$: if $T_c$
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is the child of $T$ at the bounded face $f$ of $O^{(T)}$, then
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$\varphi(T_c)$ is the child of $\varphi(T)$ at the bounded face
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$\varphi_T(f)$ of $O^{(\varphi(T))}$.
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\end{itemize}
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The outer boundaries $B_{\mathrm{out}}^{(T)}$ are \emph{not} required to
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correspond. In particular, the root tread's outer boundary may be
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degenerate (a single vertex) in $\mathcal{T}_1$ and a simple cycle in
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$\mathcal{T}_2$; this is essential because the root tread of a
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\emph{sub-tree} of $\mathcal{T}(\tilde G, \tilde S)$ inherits a
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non-degenerate $B_{\mathrm{out}}$ from its parent, even when it is iso
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to a tree arising from a single-vertex level source.
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\end{definition}
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\begin{conjecture}[Universal nesting of tire-tread trees]
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\label{conj:universal-nesting}
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Let $\mathcal{T}_2 = \mathcal{T}(G_2, S_2)$ be a tree of tire treads
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arising from a maximal planar $G_2$ with single-vertex level source
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$S_2$. Let $T \in \mathcal{T}_2$ be a tread at depth $d$, and let $f$
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be a non-trivial bounded face of $O^{(T)}$ (i.e.\ a face whose interior
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contains depth-$\ge d+2$ vertices of $G_2$). Let $\mathcal{T}_1 =
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\mathcal{T}(G_1, S_1)$ be any other tree of tire treads.
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Then there exists a maximal planar graph $\tilde G$ with single-vertex
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level source $\tilde S$ such that:
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\begin{itemize}
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\item[(N1)] $\mathcal{T}(\tilde G, \tilde S)$ contains, as a rooted
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sub-tree, an iso copy (in the sense of
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Definition~\ref{def:tree-iso-O-preserved}) of the
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truncation $\mathcal{T}_2 \setminus \mathrm{Desc}(T, f)$
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obtained from $\mathcal{T}_2$ by deleting the descendant
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sub-tree of $T$ at face $f$;
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\item[(N2)] the sub-tree of $\mathcal{T}(\tilde G, \tilde S)$ rooted at
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the (new) child of $T$'s image at (the image of) $f$ is iso,
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in the sense of Definition~\ref{def:tree-iso-O-preserved},
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to $\mathcal{T}_1$.
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\end{itemize}
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Informally: trees of tire treads are closed under face-slot insertion,
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where the slot at face $f$ in $\mathcal{T}_2$ is filled by the entirety
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of $\mathcal{T}_1$. The class of trees of tire treads is
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\emph{closed under composition} by face-slot insertion.
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\end{conjecture}
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\begin{conjecture}[Seam realizability; technical core of nesting]
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\label{conj:seam-realizability}
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Let $\mathcal{T}_1 = \mathcal{T}(G_1, S_1)$ be a tree of tire treads.
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For every integer $k \ge 3$ there exists a planar graph $H_k$, embedded
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in a closed disk $D \subset \mathbb{R}^2$ with $\partial D$ a $k$-cycle,
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such that:
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\begin{itemize}
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\item[(S1)] $\partial H_k = \partial D$, as a cyclic sequence of $k$
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vertices;
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\item[(S2)] every bounded face of $H_k$ is a triangle;
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\item[(S3)] BFS in $H_k$ from the cycle $\partial H_k$ assigns levels to
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$V(H_k) \setminus V(\partial H_k)$, and the resulting rooted
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tree of tire treads --- with the depth-$0$ tread taking
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outer boundary $\partial H_k$ in place of a single-vertex
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source --- is iso, in the sense of
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Definition~\ref{def:tree-iso-O-preserved}, to
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$\mathcal{T}_1$.
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\end{itemize}
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The construction $\mathcal{T}(H_k, \partial H_k)$ is the natural
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extension of Theorem~\ref{thm:tread-tree} from single-vertex sources to
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cycle sources: the depth-$0$ tread has non-degenerate
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$B_{\mathrm{out}} = \partial H_k$ and the rest of the construction is
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unchanged.
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\end{conjecture}
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.95\textwidth]{fig_seam_construction.png}
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\caption{Seam realizability for a small example. $(a)$ A stacked-ring
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triangulation $G_1$ with single-vertex source $S_1 = \{0\}$ and concentric
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levels $L_1, L_2, L_3$; its tree of tire treads is the chain $T_0 \to T_1
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\to T_2$ with $O^{(T_d)} = G_1[L_{d+1}]$ a $3$-cycle on each tread.
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$(b)$ The apex-removal seam construction
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$H_5 = (G_1 \setminus S_1) \cup A_5$, re-embedded so that the former
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fan-face around $S_1$ becomes the outer face (with $L_1$ now the
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outermost $G_1$-derived ring and $L_3$ innermost), and with an annular
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triangulation $A_5$ (orange) attaching to a fresh $5$-cycle $\partial H_5$
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(red). Vertex labels show $\mathrm{BFS}_{\partial H_5}$ levels in $H_5$:
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they agree with $\ell_{G_1}$ on $V(G_1) \setminus \{S_1\}$, so
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$\mathcal{T}(H_5, \partial H_5)$ is iso (combinatorial, $O$-preserved) to
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$\mathcal{T}(G_1, S_1)$.}
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\label{fig:seam-construction}
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\end{figure}
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\begin{remark}[Nesting reduces to seam realizability]
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\label{rem:seam-reduces-nesting}
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Conjecture~\ref{conj:universal-nesting} follows from
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Conjecture~\ref{conj:seam-realizability} by a direct gluing argument
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within the framework of this paper. Briefly: given a disk realization
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$H_k$ of $\mathcal{T}_1$ with $k = |C_f|$, where $C_f$ is the cycle
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bounding $f$ in $O^{(T)}$, excise from $G_2$ all vertices and edges
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strictly inside $f$, then glue $H_k$ into the resulting hole by
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identifying $\partial H_k$ with $C_f$. The verification that the glued
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graph $\tilde G$ is maximal planar, retains $\tilde S = S_2$ as a
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single-vertex level source, and realizes the claimed nesting --- the
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levels of $\tilde G$ from $S_2$ inside $f$ being just BFS-from-$C_f$ in
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$H_k$ shifted by $d + 1$ --- is mechanical from
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Theorems~\ref{thm:tread-partition},
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\ref{thm:inner-dual-outerplanar},
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\ref{thm:tread-tree} and the parent--child interface description of
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Remark~\ref{rem:tree-coloring-factorisation}.
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The substantive content of universal nesting thus sits entirely in
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Conjecture~\ref{conj:seam-realizability}: given an arbitrary tree of
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tire treads, can it be realized as the BFS-from-boundary tree of treads
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of a triangulated planar disk, for every boundary length $k \ge 3$?
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\end{remark}
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\begin{remark}[Motivation and open questions]
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\label{rem:nesting-motivation}
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The conjectured closure under nesting carries two structural
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implications for the Four Colour Theorem programme:
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\begin{itemize}
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\item \emph{Compositional colourability.} If colourability of
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$\tilde G$ in (N1)--(N2) can be decided from the colourability
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of $G_1$ and $G_2$ alone (via the parent--child consistency
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constraints of Remark~\ref{rem:tree-coloring-factorisation}),
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then $4$-colourability propagates through nesting. A minimum
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$4$CT counterexample (if it exists) would have to be
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\emph{irreducible} under such nesting --- it could not be
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decomposed into strictly smaller trees of tire treads whose
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colourings combine to a colouring of the whole.
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\item \emph{Universality.} Universal nesting positions trees of
|
||||
tire treads as a kind of ``term algebra'' for the structural
|
||||
decomposition of plane triangulations. Coloring arguments
|
||||
can then be formulated inductively on this term algebra,
|
||||
with the chain-pigeonhole step
|
||||
(Remark~\ref{rem:tree-coloring-factorisation}) supplying the
|
||||
composition rule.
|
||||
\end{itemize}
|
||||
|
||||
\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}
|
||||
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{tait-original}
|
||||
|
||||
Reference in New Issue
Block a user