diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 48210bc..ab4e642 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -30,6 +30,9 @@ \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{conj:universal-nesting}{{1.20}{12}} +\newlabel{rem:nesting-motivation}{{1.21}{12}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -38,6 +41,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } -\gdef \@abspage@last{12} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{13}{}\protected@file@percent } +\gdef \@abspage@last{13} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index d2f4a7e..64faef6 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 03:04 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,45 +511,45 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. 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PDF statistics: - 177 PDF objects out of 1000 (max. 8388607) - 107 compressed objects within 2 object streams + 181 PDF objects out of 1000 (max. 8388607) + 110 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) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 5045339..0c7a22c 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 8e1df3b..81e9dac 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -950,6 +950,75 @@ 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{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} +\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 +\emph{closed under composition} by face-slot insertion. +\end{conjecture} + +\begin{remark} +\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} + +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? +\end{remark} + \begin{thebibliography}{9}