diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux index 1122386..fc8149c 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -25,19 +25,22 @@ \newlabel{rem:no-interior-constraint}{{3.2}{3}} \newlabel{def:boundary-sequences}{{3.3}{3}} \newlabel{def:heawood-compatible}{{3.4}{3}} +\newlabel{rem:compat-is-heawood}{{3.5}{4}} +\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Why the programme runs between nested clusters}}{4}{}\protected@file@percent } +\newlabel{prop:two-sided-decomposition}{{3.6}{4}} \bibcite{Heawood1898}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} \bibcite{bauerfeld-nested-tire-duals}{5} \newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:compat-is-heawood}{{3.5}{4}} -\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}} -\newlabel{conj:heawood-chain-pigeonhole}{{3.6}{4}} -\newlabel{conj:heawood-route-fct}{{3.7}{4}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } -\gdef \@abspage@last{4} +\newlabel{rem:why-clusters}{{3.7}{5}} +\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} +\newlabel{conj:heawood-route-fct}{{3.9}{5}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log index e374fe4..b30ce1f 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/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) 17 JUN 2026 01:03 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 01:10 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,39 +192,40 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2] [3] [4] (./paper.aux) ) +[2] [3] [4] [5] (./paper.aux) ) Here is how much of TeX's memory you used: - 3012 strings out of 478268 - 42080 string characters out of 5846347 - 342271 words of memory out of 5000000 - 21059 multiletter control sequences out of 15000+600000 - 476880 words of font info for 57 fonts, out of 8000000 for 9000 + 3017 strings out of 478268 + 42161 string characters out of 5846347 + 342292 words of memory out of 5000000 + 21063 multiletter control sequences out of 15000+600000 + 477578 words of font info for 59 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,242b,264s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (4 pages, 241965 bytes). + 69i,7n,76p,242b,272s stack positions out of 10000i,1000n,20000p,200000b,200000s + + +Output written on paper.pdf (5 pages, 258014 bytes). PDF statistics: - 112 PDF objects out of 1000 (max. 8388607) - 67 compressed objects within 1 object stream + 120 PDF objects out of 1000 (max. 8388607) + 72 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf index fed3346..1e7c7e7 100644 Binary files a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf and b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf differ diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex index 21e351b..9c5ae67 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -303,34 +303,130 @@ $\{+1,-1\}$ face-labelling of $G$ satisfying proper $4$-vertex-colouring of $G$. \end{remark} +\subsection*{Why the programme runs between nested clusters} + +The vanishing condition \eqref{eq:heawood-face-sum-dual} at a vertex $v$ +is a constraint on the \emph{full} face-star of $v$. To run a +pigeonhole between two objects --- a child and a parent --- we need that +full sum to split as exactly two one-sided contributions, so that each +vertex label is the combination of a single child value and a single +parent value. This is true at the level of connected tire clusters, and +\emph{false} at the level of individual tires. Extend a Heawood +face-labelling to a connected tire cluster $\mathsf{K}$ by labelling +every annular face of every tire of $\mathsf{K}$, and for $v \in +V(\mathsf{K})$ write +\[ + \lambda^{\!*}_{\mathsf{K}}(v) \;:=\; + \sum_{f} \lambda(f) \;\bmod 3, +\] +the sum over the annular faces of $\mathsf{K}$ incident to $v$. + +\begin{proposition}[Two-sided cluster decomposition at a vertex] +\label{prop:two-sided-decomposition} +Let $v \in V(G)$ have level $\ell = \ell_G(v)$, and let +$\mathsf{K}_{\ell}$ and $\mathsf{K}_{\ell-1}$ be the at most two +connected tire clusters containing $v$, of depths $\ell$ and $\ell-1$ +respectively (Proposition~\ref{prop:two-clusters-per-vertex}). Then the +bounded faces of $G$ incident to $v$ partition into the annular faces of +$\mathsf{K}_{\ell}$ at $v$ and the annular faces of $\mathsf{K}_{\ell-1}$ +at $v$, and +\[ + \sum_{f \ni v} \lambda(f) + \;\equiv\; + \lambda^{\!*}_{\mathsf{K}_{\ell}}(v) + + \lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v) + \pmod 3 . +\] +Each one-sided value $\lambda^{\!*}_{\mathsf{K}_d}(v)$ is the +\emph{complete} sum over all depth-$d$ faces at $v$, so the Heawood +condition \eqref{eq:heawood-face-sum-dual} at $v$ reads +\[ + \lambda^{\!*}_{\mathsf{K}_{\ell}}(v) + + \lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v) \;\equiv\; 0 \pmod 3 , +\] +a pairing between the single child cluster $\mathsf{K}_{\ell}$ and the +single parent cluster $\mathsf{K}_{\ell-1}$. (When $\ell = 0$, or when +$v$ bounds no depth-$\ell$ face, only one term is present.) +\end{proposition} + +\begin{proof} +By Proposition~\ref{prop:two-clusters-per-vertex} (Step~1) every bounded +face incident to $v$ has depth $\ell-1$ or $\ell$, partitioning the +incident faces by depth; by Step~2 all depth-$\ell$ faces at $v$ lie in +the single cluster $\mathsf{K}_{\ell}$ and all depth-$(\ell-1)$ faces at +$v$ in $\mathsf{K}_{\ell-1}$. Hence the depth-$\ell$ part is exactly the +annular faces of $\mathsf{K}_{\ell}$ at $v$, the depth-$(\ell-1)$ part +those of $\mathsf{K}_{\ell-1}$, and summing $\lambda$ over the two parts +gives the identity; \eqref{eq:heawood-face-sum-dual} is its vanishing. +\end{proof} + +\begin{remark}[Failure at the tire level] +\label{rem:why-clusters} +Proposition~\ref{prop:two-sided-decomposition} is what makes the binary +parent/child pairing possible, and it requires the cluster. A vertex +$v$ may lie on many depth-$\ell$ tires --- the unbounded case of +Section~\ref{sec:tire-clusters} --- and the per-tire value +$\lambda^{\!*}(v)$ of Definition~\ref{def:heawood-labelling} then records +only the faces of \emph{one} tire at $v$, a fragment of $v$'s face-star. +No single child tire carries the complete depth-$\ell$ sum, so the label +$\sum_{f \ni v}\lambda(f)$ cannot be written as one child value plus one +parent value, and per-tire compatibility +(Definition~\ref{def:heawood-compatible}) fails to assemble to +\eqref{eq:heawood-face-sum-dual}. Clustering repairs this: +Proposition~\ref{prop:two-clusters-per-vertex} guarantees exactly one +cluster meets $v$ on each side, so $\lambda^{\!*}_{\mathsf{K}_{\ell}}(v)$ +is the complete child contribution and +$\lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v)$ the complete parent +contribution. Every vertex label is then realised as the combination of +a single child-cluster value with a single parent-cluster value, and the +pigeonhole programme below chains \emph{nested connected tire clusters} +rather than individual tires. +\end{remark} + +We write $R_{\mathsf{K}}$ for the \emph{cluster Heawood restriction +relation}: the set of (outer, inner) boundary Heawood sequence pairs +realisable by a face-labelling of $\mathsf{K}$, defined as in +Definition~\ref{def:boundary-sequences} but with the outer and inner +boundaries of the cluster and the complete one-sided values +$\lambda^{\!*}_{\mathsf{K}}$ in place of a single tire's, read up to +rotation and global sign-flip. By +Proposition~\ref{prop:two-sided-decomposition} two nested clusters are +compatible along their shared interface exactly when the inner sequence +of the parent is the pointwise negation mod $3$ of the outer sequence of +the child (after the orientation reversal of +Definition~\ref{def:heawood-compatible}). + \begin{conjecture}[Heawood chain-pigeonhole principle] \label{conj:heawood-chain-pigeonhole} There is a function $N(k)$ such that the following holds. Let \[ - T_0 \supset T_1 \supset \cdots \supset T_{N(k)} + \mathsf{K}_0 \supset \mathsf{K}_1 \supset \cdots \supset + \mathsf{K}_{N(k)} \] -be a nested chain of tires in $\mathcal{T}(G, S)$ whose shared interface -cycles have length at most $k$. Then two adjacent Heawood restriction -relations $R_{T_i}, R_{T_{i+1}}$ in the chain admit compatible -face-labellings along their shared interface -(Definition~\ref{def:heawood-compatible}), after rotation and global -sign-flip. Equivalently, the chain contains a local gluing step that -cannot be obstructed by disjoint Heawood boundary restrictions. +be a nested chain of connected tire clusters in $\mathcal{T}(G, S)$ whose +shared interfaces have length at most $k$. Then two adjacent cluster +restriction relations $R_{\mathsf{K}_i}, R_{\mathsf{K}_{i+1}}$ in the +chain admit compatible face-labellings along their shared interface, +after rotation and global sign-flip. Equivalently, the chain contains a +local gluing step that cannot be obstructed by disjoint Heawood boundary +restrictions. \end{conjecture} -\begin{conjecture}[Heawood tire route to the Four Colour Theorem] +\begin{conjecture}[Heawood cluster route to the Four Colour Theorem] \label{conj:heawood-route-fct} For every plane triangulation $G$ and every level source $S$, the -Heawood restriction relations $\{R_T : T \in \mathcal{T}(G, S)\}$ admit -a selection of face-labellings that is compatible along every interface -of the tire tree. By Remark~\ref{rem:compat-is-heawood} this yields a -$\{+1,-1\}$ face-labelling of $G$ satisfying -\eqref{eq:heawood-face-sum-dual}, hence $G$ is properly -$4$-vertex-colourable. +cluster Heawood restriction relations +$\{R_{\mathsf{K}} : \mathsf{K} \text{ a connected tire cluster}\}$ admit +a selection of face-labellings that is compatible along every cluster +interface. By Proposition~\ref{prop:two-sided-decomposition} and +Remark~\ref{rem:compat-is-heawood} this yields a $\{+1,-1\}$ +face-labelling of $G$ satisfying \eqref{eq:heawood-face-sum-dual}, hence +$G$ is properly $4$-vertex-colourable. \end{conjecture} -%% TODO: realisability of $R_T$ per tire; counting / pigeonhole bound -%% giving $N(k)$; orientation/reversal bookkeeping on $\gamma$. +%% TODO: realisability of $R_{\mathsf{K}}$ per cluster; counting / +%% pigeonhole bound giving $N(k)$; orientation/reversal bookkeeping on +%% the shared interface. \begin{thebibliography}{9}