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 315956b..43d4d2b 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -5,17 +5,30 @@ \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} +\citation{Heawood1898} +\citation{bauerfeld-medial-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Heawood restrictions on the tire dual}}{2}{}\protected@file@percent } +\newlabel{sec:heawood-restrictions}{{2}{2}} +\newlabel{def:heawood-labelling}{{2.1}{2}} +\newlabel{rem:no-interior-constraint}{{2.2}{2}} +\newlabel{def:boundary-sequences}{{2.3}{2}} +\newlabel{def:heawood-compatible}{{2.4}{2}} \bibcite{Heawood1898}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} -\bibcite{bauerfeld-nested-tire-duals}{4} +\bibcite{bauerfeld-medial-tires}{4} +\bibcite{bauerfeld-nested-tire-duals}{5} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Heawood restrictions on the tire dual}}{1}{}\protected@file@percent } -\newlabel{sec:heawood-restrictions}{{2}{1}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{1}{}\protected@file@percent } -\gdef \@abspage@last{1} +\newlabel{rem:compat-is-heawood}{{2.5}{3}} +\newlabel{eq:heawood-face-sum-dual}{{2.1}{3}} +\newlabel{conj:heawood-chain-pigeonhole}{{2.6}{3}} +\newlabel{conj:heawood-route-fct}{{2.7}{3}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } +\gdef \@abspage@last{3} 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 f5482e0..a8bd287 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 00:31 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 00:42 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,31 +192,35 @@ 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}] -(./paper.aux) ) +[2] [3] (./paper.aux) ) Here is how much of TeX's memory you used: - 2992 strings out of 478268 - 41657 string characters out of 5846347 - 336140 words of memory out of 5000000 - 21041 multiletter control sequences out of 15000+600000 + 3002 strings out of 478268 + 41895 string characters out of 5846347 + 340223 words of memory out of 5000000 + 21051 multiletter control sequences out of 15000+600000 475975 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,5n,76p,242b,213s stack positions out of 10000i,1000n,20000p,200000b,200000s - + 69i,7n,76p,242b,278s stack positions out of 10000i,1000n,20000p,200000b,200000s + -Output written on paper.pdf (1 page, 145461 bytes). +Output written on paper.pdf (3 pages, 210205 bytes). PDF statistics: - 68 PDF objects out of 1000 (max. 8388607) - 40 compressed objects within 1 object stream + 94 PDF objects out of 1000 (max. 8388607) + 56 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 1eac855..0e01271 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 a670454..882182a 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -87,14 +87,149 @@ Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ and $G$ has $2n - 4$ triangular faces. -%% TODO: state the Heawood restriction this paper studies. The relevant -%% classical input is Heawood's face-sum identity \cite{Heawood1898}; the goal -%% here is to record what it forces on the dual of a (nested) tire graph. +The classical input is Heawood's face-sum identity \cite{Heawood1898}: +for any proper $3$-edge-colouring of a cubic plane graph $H$, assigning +each face of $H$ a number in $\{+1, -1\}$ can be done so that the labels +around every vertex of $H$ sum to $0 \pmod 3$. In the triangulation +$G$ dual to $H$ this becomes a $\{+1, -1\}$ labelling of the +\emph{faces} of $G$ whose incident-face sum at every vertex of $G$ +vanishes mod $3$. Our aim is to record what this restriction forces +along the boundary cycles of a nested tire graph, and to formulate a +chain-pigeonhole programme in this Heawood labelling parallel to the +medial programme of \cite{bauerfeld-medial-tires}. \section{Heawood restrictions on the tire dual} \label{sec:heawood-restrictions} -%% TODO: main development. +We work inside a fixed nested tire decomposition $\mathcal{T}(G, S)$ of +$G$ from a single-vertex level source $S$ \cite{bauerfeld-nested-tires}, +and use the tire data $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ with +annular faces $F_{\mathrm{ann}}$, outer boundary $B_{\mathrm{out}}$, and +inner boundary $B_{\mathrm{in}}$ +(\cite[Definition~1.5]{bauerfeld-nested-tires}). Since $O$ is +outerplanar, every vertex of a tire lies on $B_{\mathrm{out}}$ or on the +inner-boundary walk $B_{\mathrm{in}}$; a tire has no interior vertices. + +\begin{definition}[Heawood face-labelling of a tire] +\label{def:heawood-labelling} +A \emph{Heawood face-labelling} of a tire graph $T$ is a map +\[ + \lambda : F_{\mathrm{ann}} \longrightarrow \{+1, -1\} +\] +assigning a sign to each annular face of $T$. For a vertex +$v \in V(T)$, write $F_{\mathrm{ann}}(v) \subseteq F_{\mathrm{ann}}$ for +the set of annular faces of $T$ incident to $v$, and define the +\emph{induced vertex value} +\[ + \lambda^{\!*}(v) \;:=\; \sum_{f \in F_{\mathrm{ann}}(v)} \lambda(f) + \;\;\bmod 3 \;\in\; \{0, 1, -1\}. +\] +The value $\lambda^{\!*}(v)$ is the \emph{partial} face-sum at $v$ taken +over the annular faces of $T$ alone, not over all faces of $G$ incident +to $v$. +\end{definition} + +\begin{remark} +\label{rem:no-interior-constraint} +Because a tire has no interior vertices, every annular face of $T$ is +incident to $B_{\mathrm{out}} \cup B_{\mathrm{in}}$, and a Heawood +face-labelling is subject to \emph{no} internal constraint: all +$2^{|F_{\mathrm{ann}}|}$ sign assignments are admissible. The Heawood +restriction is felt only on the two boundary cycles, through the induced +vertex values $\lambda^{\!*}$. +\end{remark} + +\begin{definition}[Induced boundary sequences] +\label{def:boundary-sequences} +Let $\lambda$ be a Heawood face-labelling of $T$. Reading the vertices +of $B_{\mathrm{out}}$ in clockwise order $v_0, v_1, \dots, v_{p-1}$, the +\emph{outer Heawood sequence} of $(T, \lambda)$ is +\[ + \sigma_{\mathrm{out}}(T, \lambda) + \;:=\; \bigl(\lambda^{\!*}(v_0), \dots, \lambda^{\!*}(v_{p-1})\bigr) + \;\in\; \{0, 1, -1\}^{p}. +\] +Reading the inner-boundary walk $B_{\mathrm{in}}$ in clockwise order +$w_0, \dots, w_{q-1}$ gives the \emph{inner Heawood sequence} +$\sigma_{\mathrm{in}}(T, \lambda) \in \{0, 1, -1\}^{q}$. The +\emph{Heawood restriction relation} of $T$ is the set +\[ + R_T \;:=\; \bigl\{\, + \bigl(\sigma_{\mathrm{out}}(T, \lambda),\, + \sigma_{\mathrm{in}}(T, \lambda)\bigr) + \;:\; \lambda : F_{\mathrm{ann}} \to \{+1, -1\} + \,\bigr\} +\] +of all (outer, inner) sequence pairs realisable by a single +face-labelling, read up to rotation and the global sign-flip +$\lambda \mapsto -\lambda$ (equivalently +$\sigma \mapsto -\sigma$). +\end{definition} + +\begin{definition}[Heawood compatibility across an interface] +\label{def:heawood-compatible} +Let $T$ be a tire and $T' \in \mathcal{T}(G, S)$ a child of $T$, so the +outer boundary cycle $B_{\mathrm{out}}^{(T')}$ coincides with a bounded +face of $O^{(T)}$; let $\gamma$ be this shared cycle, of length $L$, and +let $v$ range over its vertices. Heawood face-labellings $\lambda$ of +$T$ and $\lambda'$ of $T'$ are \emph{compatible along $\gamma$} if at +every shared vertex $v$, +\[ + \lambda^{\!*}(v) + (\lambda')^{\!*}(v) \;\equiv\; 0 \pmod 3, +\] +i.e.\ $0$ is paired with $0$ and $+1$ with $-1$. Equivalently, the +inner Heawood sequence of $T$ on $\gamma$ is the pointwise negation +mod $3$ of the outer Heawood sequence of $T'$ on $\gamma$, after +reversing one of the two clockwise readings to account for the opposite +rotational senses in which $T$ and $T'$ traverse $\gamma$. +\end{definition} + +\begin{remark} +\label{rem:compat-is-heawood} +Compatibility along $\gamma$ at $v$ is exactly the statement that the +full incident-face sum at $v$ --- over the parent's annular faces +together with the child's --- vanishes mod $3$: +\begin{equation} +\label{eq:heawood-face-sum-dual} + \sum_{f \ni v} \lambda(f) \;\equiv\; 0 \pmod 3 + \qquad\text{for every vertex } v \in V(G). +\end{equation} +Since $\gamma$ carries all faces of $G$ incident to $v$ between the two +tires, a family of Heawood face-labellings that is pairwise compatible +along every interface of $\mathcal{T}(G, S)$ assembles into a single +$\{+1,-1\}$ face-labelling of $G$ satisfying +\eqref{eq:heawood-face-sum-dual} at every vertex, hence (by Tait) a +proper $4$-vertex-colouring of $G$. +\end{remark} + +\begin{conjecture}[Heawood chain-pigeonhole principle] +\label{conj:heawood-chain-pigeonhole} +There is a function $N(k)$ such that the following holds. Let +\[ + T_0 \supset T_1 \supset \cdots \supset T_{N(k)} +\] +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. +\end{conjecture} + +\begin{conjecture}[Heawood tire 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. +\end{conjecture} + +%% TODO: realisability of $R_T$ per tire; counting / pigeonhole bound +%% giving $N(k)$; orientation/reversal bookkeeping on $\gamma$. \begin{thebibliography}{9} @@ -113,6 +248,11 @@ E.~Bauerfeld, \emph{Nested Tire Decompositions of Plane Triangulations}, manuscript (math-research repository), 2026. +\bibitem{bauerfeld-medial-tires} +E.~Bauerfeld, +\emph{Medial Tire Decompositions of Plane Triangulations}, +manuscript (math-research repository), 2026. + \bibitem{bauerfeld-nested-tire-duals} E.~Bauerfeld, \emph{Coloring Nested Tire Dual Graphs},