From bfa8b8956d237548a5e056c0c33b58b98ee16853 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 04:39:49 -0400 Subject: [PATCH] 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 --- papers/coloring_nested_tire_graphs/paper.aux | 13 +- papers/coloring_nested_tire_graphs/paper.log | 91 ++++----- papers/coloring_nested_tire_graphs/paper.pdf | Bin 927914 -> 618557 bytes papers/coloring_nested_tire_graphs/paper.tex | 190 ------------------- 4 files changed, 39 insertions(+), 255 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 4a9e157..48210bc 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -30,13 +30,6 @@ \newlabel{rem:count-general-outerplanar}{{1.16}{10}} \newlabel{thm:tread-tree}{{1.17}{10}} \newlabel{rem:tree-multiple-children}{{1.18}{11}} -\newlabel{rem:tree-coloring-factorisation}{{1.19}{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} @@ -45,6 +38,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:nesting-motivation}{{1.24}{14}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent } -\gdef \@abspage@last{14} +\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } +\gdef \@abspage@last{12} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index b7995d2..667ad80 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 04:24 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 04:39 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,64 +511,45 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. 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[]\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) ) +[7] [8] [9] [10] [11] [12] (./paper.aux) ) Here is how much of TeX's memory you used: - 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 + 14048 strings out of 478268 + 279229 string characters out of 5846347 + 563840 words of memory out of 5000000 + 31872 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 (14 pages, 927914 bytes). + + +Output written on paper.pdf (12 pages, 618557 bytes). 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-\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 $\varphi : \mathcal{T}_1 \to \mathcal{T}_2$ 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} - -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{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: -\begin{itemize} -\item \emph{Compositional colourability.} If colourability of - $\tilde G$ 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 - \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. -\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}