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 15db1e2..7ff7c88 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -30,10 +30,11 @@ \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} +\citation{bauerfeld-nested-tires} \newlabel{rem:why-clusters}{{3.7}{5}} \newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} \newlabel{conj:heawood-route-fct}{{3.9}{5}} +\bibcite{Heawood1898}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} @@ -43,5 +44,10 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The constraint floor}}{6}{}\protected@file@percent } +\newlabel{sec:constraint-floor}{{4}{6}} +\newlabel{def:achievable-boundary-set}{{4.1}{6}} +\newlabel{prop:constraint-floor}{{4.2}{6}} +\newlabel{rem:floor-consequences}{{4.3}{6}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } \gdef \@abspage@last{6} 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 345706b..167cd74 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 01:32 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 02:14 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,37 +192,47 @@ 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] [5] [6] (./paper.aux) ) +[2] [3] [4] [5] +Overfull \hbox (20.41568pt too wide) detected at line 494 +\OML/cmm/m/it/10 ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) = +\OML/cmm/m/it/10 ^^U[]; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n +/10 ) = \OML/cmm/m/it/10 ^^U[] \OT1/cmr/m/n/10 + \OML/cmm/m/it/10 ^^U[] \OT1/c +mr/m/n/10 (1 \OML/cmm/m/it/10 < i < n \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 1)\O +ML/cmm/m/it/10 ; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) = + \OML/cmm/m/it/10 ^^U[]; ^^U[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/ +n/10 ) = [][] \OML/cmm/m/it/10 ^^U[]: + [] + +[6] (./paper.aux) ) Here is how much of TeX's memory you used: - 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 + 3021 strings out of 478268 + 42259 string characters out of 5846347 + 342330 words of memory out of 5000000 + 21067 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,340s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (6 pages, 259543 bytes). + 69i,7n,76p,242b,290s stack positions out of 10000i,1000n,20000p,200000b,200000s +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmss10.pfb> +Output written on paper.pdf (6 pages, 266448 bytes). PDF statistics: 123 PDF objects out of 1000 (max. 8388607) 74 compressed objects within 1 object stream 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 cb02559..02545ce 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 372c89c..0a121ab 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -441,6 +441,84 @@ $G$ is properly $4$-vertex-colourable. %% pigeonhole bound giving $N(k)$; orientation/reversal bookkeeping on %% the shared interface. +\section{The constraint floor} +\label{sec:constraint-floor} + +A nested substructure constrains its outer interface through the set of +Heawood boundary sequences it can realise. By the self-similarity of the +tire decomposition (\cite{bauerfeld-nested-tires}), the region $G_T$ +enclosed by a tire's outer cycle, away from the source, is itself a +triangulated disk; we record how tightly any such disk can constrain its +boundary. The bound below depends only on the disk triangulation, not on +a tire-tree labelling. + +\begin{definition}[Achievable boundary set of a disk] +\label{def:achievable-boundary-set} +Let $D$ be a triangulated disk whose boundary is a simple $n$-cycle +$C = (v_0, \dots, v_{n-1})$. Call a Heawood face-labelling +$\lambda : F(D) \to \{+1,-1\}$ \emph{interior-valid} if +$\sum_{f \ni w} \lambda(f) \equiv 0 \pmod 3$ at every interior vertex $w$ +of $D$ (no condition on $C$). The \emph{achievable boundary set} of $D$ +is +\[ + \Phi(D) \;:=\; \bigl\{\, + (\lambda^{*}(v_0), \dots, \lambda^{*}(v_{n-1})) + \;:\; \lambda \text{ interior-valid} \,\bigr\} + \;\subseteq\; \{0,1,-1\}^{n} . +\] +\end{definition} + +\begin{proposition}[Constraint floor] +\label{prop:constraint-floor} +For every triangulated disk $D$ with boundary an $n$-cycle, +\[ + |\Phi(D)| \;\ge\; 2^{\,n-2}, +\] +and the bound is attained --- already by the triangulation of the +$n$-gon with no interior vertices. Consequently no nested structure +constrains the outer cycle below $2^{\,n-2}$ achievable Heawood +sequences; the trivial tire is already maximally constraining. +\end{proposition} + +\begin{proof}[Proof of attainment] +Triangulate the $n$-gon as a fan from $v_0$, with faces +$\{v_0, v_i, v_{i+1}\}$ for $1 \le i \le n-2$ and labels +$\lambda_i := \lambda(\{v_0, v_i, v_{i+1}\})$; there are no interior +vertices, so every labelling is interior-valid. The induced boundary +values are +\[ + \lambda^{*}(v_1) = \lambda_1, \quad + \lambda^{*}(v_i) = \lambda_{i-1} + \lambda_i \ \ (1 < i < n-1), \quad + \lambda^{*}(v_{n-1}) = \lambda_{n-2}, \quad + \lambda^{*}(v_0) = \textstyle\sum_{j} \lambda_j . +\] +From $\lambda^{*}(v_1)$ and the relations +$\lambda_i = \lambda^{*}(v_i) - \lambda_{i-1}$ the tuple +$(\lambda_1, \dots, \lambda_{n-2}) \in \{+1,-1\}^{n-2}$ is recovered from +the boundary sequence, so the map $\lambda \mapsto \lambda^{*}|_C$ is +injective and $|\Phi(D)| = 2^{\,n-2}$. +\end{proof} + +%% TODO (lower bound): show |Phi(D)| >= 2^{n-2} for EVERY triangulated +%% disk D. Strategy: the n boundary-incident faces (one per boundary edge) +%% carry n-2 independent binary degrees of freedom after the interior +%% Heawood constraints are imposed; those constraints relate only +%% interior-incident faces and cannot collapse the boundary freedom below +%% 2^{n-2}. (See notes/boundary_restriction_structure.tex.) + +\begin{remark} +\label{rem:floor-consequences} +Two consequences. First, $\Phi(D)$ is a $\mathbb{Z}/3$ zonotope --- a +projected cube, sign-closed but not a $\mathrm{GF}(3)$ subspace --- and at +the floor it has size $2^{\,n-2}$ with affine hull of dimension $n-2$. +Second, since the floor is exponential in the interface length $n$, a +maximally-constraining child still offers $2^{\,n-2}$ outer options, so +the gluing of Conjecture~\ref{conj:heawood-chain-pigeonhole} has the least +slack at \emph{short} interfaces (e.g.\ $n = 4$ leaves $4$ options) and is +easy at long ones; the difficulty of the programme is concentrated at +short level cycles. +\end{remark} + \begin{thebibliography}{9} \bibitem{Heawood1898}