From 8b6c2b621cb10226752ef7fddde89ba1ada6bf2c Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 13:49:44 -0400 Subject: [PATCH] split: extract foundational depth material into new plane_depth paper MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Splits the existing plane_depth_sequencing paper into two: papers/plane_depth/paper.tex (NEW, 4 pages): - Plane depth definition. - Level edge, up/down/neutral triangle classification. - Outerplanarity lemma (formerly Lemma 2.6 of PDS). - Deep embedding G' definition. - "Every face of G' is up or down" lemma. - Unique level edge per face; shared level edge between adjacent faces. - Quadrilateral decomposition definition with three types (shallow diamond, deep diamond, S quad). papers/plane_depth_sequencing/paper.tex (slimmed from 11 → 6 pages): - Cites plane_depth for all foundational definitions. - Keeps: slice, move definitions (anchor drop, level add, join, ring completion), move selection, termination theorem. papers/coloring_nested_tire_graphs/paper.tex: - Bibliography updated: cite bauerfeld-depth instead of bauerfeld-pds. - Two in-text references updated to cite the new outerplanarity lemma in plane_depth. Rationale: the outerplanarity / deep-embedding / quadrilateral- decomposition material is foundational and reused by multiple papers (and by the proposed level-cycle generalization). The quadrilateral-sequencing programme is one specific application. Splitting lets coloring_nested_tire_graphs cite the foundations cleanly without dragging in the sequencing machinery. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 6 +- papers/coloring_nested_tire_graphs/paper.log | 12 +- papers/coloring_nested_tire_graphs/paper.pdf | Bin 837514 -> 837047 bytes papers/coloring_nested_tire_graphs/paper.tex | 13 +- papers/plane_depth/paper.aux | 28 ++ papers/plane_depth/paper.log | 225 +++++++++ papers/plane_depth/paper.pdf | Bin 0 -> 191936 bytes papers/plane_depth/paper.tex | 309 ++++++++++++ papers/plane_depth_sequencing/paper.aux | 19 +- papers/plane_depth_sequencing/paper.log | 117 +++-- papers/plane_depth_sequencing/paper.pdf | Bin 250125 -> 231503 bytes papers/plane_depth_sequencing/paper.tex | 476 ++++++++++--------- 12 files changed, 904 insertions(+), 301 deletions(-) create mode 100644 papers/plane_depth/paper.aux create mode 100644 papers/plane_depth/paper.log create mode 100644 papers/plane_depth/paper.pdf create mode 100644 papers/plane_depth/paper.tex diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index cb921e3..a258f40 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -14,9 +14,9 @@ \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } \newlabel{fig:partial-tire-dual-bridge}{{4}{5}} \newlabel{prop:no-level-d-pinch}{{1.9}{5}} -\citation{bauerfeld-pds} +\citation{bauerfeld-depth} \newlabel{lem:tire-component}{{1.10}{6}} -\citation{bauerfeld-pds} +\citation{bauerfeld-depth} \newlabel{rem:tire-component-degenerate}{{1.11}{8}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} \newlabel{prop:edge-vertex-bijection}{{1.13}{8}} @@ -29,7 +29,7 @@ \newlabel{fig:facial-dual-choices}{{5}{10}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{10}{}\protected@file@percent } \newlabel{sec:latin-conjecture}{{2}{10}} -\bibcite{bauerfeld-pds}{1} +\bibcite{bauerfeld-depth}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index f759a4b..8a5cee0 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) 26 MAY 2026 03:16 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 13:49 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -226,17 +226,17 @@ LaTeX Warning: `h' float specifier changed to `ht'. [4 <./fig_partial_tire_dual.png>] [5 <./fig_partial_tire_dual_bridge.png>] [6] [7] [8] -LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 576. +LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 577. [9] File: notes/fig_facial_dual_choices.png Graphic file (type png) Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -655. +656. (pdftex.def) Requested size: 360.0pt x 143.50418pt. -Overfull \hbox (68.454pt too wide) detected at line 706 +Overfull \hbox (68.454pt too wide) detected at line 707 \OMS/cmsy/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B[]; O\OT1/cmr/m/n/10 ) := [] \OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ( \OML/cmm/m/it/10 B[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! f\OT1/cmr/m/n/10 1\OML @@ -254,7 +254,7 @@ LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 3052 strings out of 478268 - 43205 string characters out of 5846347 + 43207 string characters out of 5846347 344356 words of memory out of 5000000 21094 multiletter control sequences out of 15000+600000 475834 words of font info for 54 fonts, out of 8000000 for 9000 @@ -277,7 +277,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (11 pages, 837514 bytes). +Output written on paper.pdf (11 pages, 837047 bytes). 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bb385d0..cf2b8a0 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -384,8 +384,9 @@ the faces of $G$ in $F_{C'}$. \begin{proof} \emph{Outerplanarity of the two level parts.} By construction $S$ -lies on the outer face of $\Pi_G$, so Lemma~2.6 of \cite{bauerfeld-pds} -applies directly with $(G, \Pi_G, S)$, giving that $G[L_{d'}]$ is +lies on the outer face of $\Pi_G$, so the outerplanarity lemma of +\cite{bauerfeld-depth} applies directly with $(G, \Pi_G, S)$, giving +that $G[L_{d'}]$ is outerplanar for each $d' \geq 0$. Subgraphs of outerplanar graphs are outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are both outerplanar. @@ -447,8 +448,8 @@ $d = 0$ case) --- serves as $B_{\mathrm{out}}$. We set $B_{\mathrm{out}} := G[V_{C'} \cap L_d]$ if this is a cycle, and the single vertex $\{v_0\}$ in the degenerate case. -\emph{Inner outerplanar graph.} By Lemma~2.6 of \cite{bauerfeld-pds}, -$G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O := +\emph{Inner outerplanar graph.} By the outerplanarity lemma of +\cite{bauerfeld-depth}, $G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O := G[V_{C'} \cap L_{d+1}]$. The boundary curve(s) of $R_{C'}$ on the $L_{d+1}$ side are exactly the boundary of $O$'s outer face in the inherited embedding; this outer-face boundary is a single closed walk @@ -754,9 +755,9 @@ ongoing automated search. \begin{thebibliography}{9} -\bibitem{bauerfeld-pds} +\bibitem{bauerfeld-depth} E.~Bauerfeld, -\emph{Plane Depth Sequencing}, +\emph{Plane Depth}, manuscript (math-research repository), 2026. \end{thebibliography} diff --git a/papers/plane_depth/paper.aux b/papers/plane_depth/paper.aux new file mode 100644 index 0000000..6c7a65e --- /dev/null +++ b/papers/plane_depth/paper.aux @@ -0,0 +1,28 @@ +\relax +\citation{bauerfeld-pds-seq} +\citation{bauerfeld-nested-tires} +\citation{baker1994} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent } +\newlabel{def:plane-depth}{{1.1}{1}} +\newlabel{def:level-edge}{{1.2}{1}} +\newlabel{def:triangle-types}{{1.3}{1}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Outerplanarity of depth levels}}{2}{}\protected@file@percent } +\newlabel{lem:outerplanarity}{{2.1}{2}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Deep embedding}}{2}{}\protected@file@percent } +\newlabel{def:deep-embedding}{{3.1}{2}} +\newlabel{lem:up-down-faces}{{3.2}{2}} +\bibcite{baker1994}{1} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Quadrilateral decomposition}}{3}{}\protected@file@percent } +\newlabel{lem:unique-level-edge}{{4.1}{3}} +\newlabel{lem:shared-level-edge}{{4.2}{3}} +\newlabel{def:quad-decomposition}{{4.3}{3}} +\newlabel{def:quad-types}{{4.5}{3}} +\bibcite{bauerfeld-pds-seq}{2} +\bibcite{bauerfeld-nested-tires}{3} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git 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