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 fc8149c..15db1e2 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -25,11 +25,15 @@ \newlabel{rem:no-interior-constraint}{{3.2}{3}} \newlabel{def:boundary-sequences}{{3.3}{3}} \newlabel{def:heawood-compatible}{{3.4}{3}} +\citation{Heawood1898} \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} +\newlabel{rem:why-clusters}{{3.7}{5}} +\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}} +\newlabel{conj:heawood-route-fct}{{3.9}{5}} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-medial-tires}{4} @@ -39,8 +43,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\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} +\@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 b30ce1f..345706b 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 01:10 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,7 +192,7 @@ 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] (./paper.aux) ) +[2] [3] [4] [5] [6] (./paper.aux) ) Here is how much of TeX's memory you used: 3017 strings out of 478268 42161 string characters out of 5846347 @@ -200,32 +200,32 @@ Here is how much of TeX's memory you used: 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,272s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (5 pages, 258014 bytes). + 69i,7n,76p,242b,340s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on paper.pdf (6 pages, 259543 bytes). PDF statistics: - 120 PDF objects out of 1000 (max. 8388607) - 72 compressed objects within 1 object stream + 123 PDF objects out of 1000 (max. 8388607) + 74 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 1e7c7e7..cb02559 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 9c5ae67..372c89c 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -287,20 +287,33 @@ rotational senses in which $T$ and $T'$ traverse $\gamma$. \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 $3$: +Call $v$ \emph{interior} if it is not incident to the outer face of +$\Pi_G$. For an interior vertex every incident face is bounded, and +compatibility along $\gamma$ at $v$ is exactly the statement that the +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). + \qquad\text{for every interior 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 $G$. +the sum ranging over the bounded faces incident to $v$. The interfaces +of $\mathcal{T}(G, S)$ are interior level cycles, so cluster +compatibility only ever constrains interior vertices and is untouched by +the outer face. + +To pass from \eqref{eq:heawood-face-sum-dual} to a colouring one must +account for the outer face: an outer-boundary vertex is incident to the +unbounded face $f_\infty$, whose label is omitted from the bounded sum. +Extend $\lambda$ by a single label $\lambda(f_\infty) \in \{+1, -1\}$ on +$f_\infty$. Then a family of Heawood face-labellings that is pairwise +compatible along every interface of $\mathcal{T}(G, S)$ assembles into a +$\{+1,-1\}$ labelling of \emph{all} faces of $G$ for which +$\sum_{f \ni v} \lambda(f) \equiv 0 \pmod 3$ holds at every vertex --- +the outer-boundary vertices now carrying $\lambda(f_\infty)$ in their +sum. This is Heawood's face-sum identity \cite{Heawood1898} for a +proper $3$-edge-colouring of the full cubic dual of $G$, hence (by Tait) +a proper $4$-vertex-colouring of $G$. \end{remark} \subsection*{Why the programme runs between nested clusters}