diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 44ee562..022f824 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -5,13 +5,16 @@ \newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:tire-graph}{{1.5}{2}} \newlabel{rem:tire-counts}{{1.6}{2}} -\newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{0pt} -\newlabel{tocindent1}{17.77782pt} -\newlabel{tocindent2}{0pt} -\newlabel{tocindent3}{0pt} +\citation{bauerfeld-pds} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{3}} \newlabel{lem:tire-component}{{1.7}{3}} -\newlabel{rem:tire-component-degenerate}{{1.8}{3}} -\gdef \@abspage@last{3} +\bibcite{bauerfeld-pds}{1} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\newlabel{rem:tire-component-degenerate}{{1.8}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 4c7d95c..b6c5179 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) 25 MAY 2026 15:00 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:17 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -210,34 +210,35 @@ Package pdftex.def Info: fig_tire_example.png used on input line 154. LaTeX Warning: `h' float specifier changed to `ht'. -[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] (./paper.aux) ) +[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] [4] (./paper.aux) ) Here is how much of TeX's memory you used: - 3005 strings out of 478268 - 41970 string characters out of 5846347 - 338147 words of memory out of 5000000 - 21052 multiletter control sequences out of 15000+600000 + 3006 strings out of 478268 + 41985 string characters out of 5846347 + 339156 words of memory out of 5000000 + 21053 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,625b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (3 pages, 449333 bytes). + 69i,8n,76p,625b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on paper.pdf (4 pages, 464467 bytes). PDF statistics: - 89 PDF objects out of 1000 (max. 8388607) - 51 compressed objects within 1 object stream + 97 PDF objects out of 1000 (max. 8388607) + 56 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 11 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 ef963f7..cb273ff 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 f05d45d..f92ab08 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -194,6 +194,47 @@ $C$ inside its closed boundary region are exactly the faces of $G$ in $F_{C'}$. \end{lemma} +\begin{proof}[Proof sketch] +By Lemma~2.6 of \cite{bauerfeld-pds} (whose argument, given for an +outer-cycle source, extends verbatim to an arbitrary level source $S$ +by treating $S$ as the depth-$0$ set), the subgraph $G[L_{d'}]$ is +outerplanar for each $d' \geq 0$. Since subgraphs of outerplanar +graphs are outerplanar, both $G[V_{C'} \cap L_d]$ and +$G[V_{C'} \cap L_{d+1}]$ are outerplanar. + +Layer containment (a consequence of the bounded-step property of BFS +on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$ +has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap +L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least +one vertex in $L_d$. + +It remains to identify the boundary of $R_{C'} := \bigcup_{f \in +F_{C'}} f \subseteq |\Pi_G|$. Each edge on the topological boundary +$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a +face $f' \notin F_{C'}$. Such an $f'$ has dual depth in $\{d-1, d+1\}$: +if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a +distinct component of $G'_d$, which contradicts the connectivity of +$C'$ together with the fact that adjacent depth-$d$ dual vertices belong +to the same component. A short case analysis on the level of the third +vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have +both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both +endpoints in $L_{d+1}$. Each connected component of $\partial R_{C'}$ +is therefore monochromatic in level. + +Since $C'$ is connected and $R_{C'}$ is a connected planar 2-complex +of triangles glued along edges, $\partial R_{C'}$ consists of either +one closed walk (when $R_{C'}$ is a topological disk) or two closed +walks (when $R_{C'}$ is a topological annulus). These walks are +simple cycles in $G$ on the respective level sets (with one cycle +possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the +endpoints of the BFS, giving the degenerate-boundary case of +Definition~\ref{def:tire-graph}). Together with the outerplanarity of +$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above, +these cycles serve as the two boundary parts of $C$, in either order, +and the depth-$d$ triangles in $F_{C'}$ tile the closed region between +them. +\end{proof} + \begin{remark} \label{rem:tire-component-degenerate} Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be @@ -206,4 +247,13 @@ the orientation of the inherited embedding (equivalently, on which side of $C$ contains the rest of $\Pi_G$). \end{remark} +\begin{thebibliography}{9} + +\bibitem{bauerfeld-pds} +E.~Bauerfeld, +\emph{Plane Depth Sequencing}, +manuscript (math-research repository), 2026. + +\end{thebibliography} + \end{document}