From 927f9b7adb8b8dafa986e6e2f66204370584599e Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 25 Apr 2026 05:22:25 -0400 Subject: [PATCH] Simplify d=0 case in outerplanar lemma proof Co-Authored-By: Claude Sonnet 4.6 --- .../plane_depth_sequencing/paper.fdb_latexmk | 8 +- papers/plane_depth_sequencing/paper.log | 89 +++++------------- papers/plane_depth_sequencing/paper.pdf | Bin 158297 -> 158166 bytes .../plane_depth_sequencing/paper.synctex.gz | Bin 0 -> 19983 bytes papers/plane_depth_sequencing/paper.tex | 2 +- 5 files changed, 27 insertions(+), 72 deletions(-) create mode 100644 papers/plane_depth_sequencing/paper.synctex.gz diff --git a/papers/plane_depth_sequencing/paper.fdb_latexmk b/papers/plane_depth_sequencing/paper.fdb_latexmk index 47b7adf..adaeb37 100644 --- a/papers/plane_depth_sequencing/paper.fdb_latexmk +++ b/papers/plane_depth_sequencing/paper.fdb_latexmk @@ -1,6 +1,6 @@ # Fdb version 4 -["pdflatex"] 1777106346.98886 "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" "paper.pdf" "paper" 1777106347.2627 0 - "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" 1777106346.52452 7702 1354320d93134a233d8c3c035e09f7c0 "" +["pdflatex"] 1777106635.82554 "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" "paper.pdf" "paper" 1777106636.10525 0 + "/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex" 1777106635.61829 7628 2263cb80db1ec8aa25bb60c5652b01c2 "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1 5523663 ec1f96d89b308e150332b305019a3402 "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/pdftex/pdflatex.fmt" 1 3600504 177ced77725200f4fa24b79427ded12f "" "/nix/store/4g7bv3lsd1r7lrfxi0x145xac0jag4hl-texlive-combined-full-2025.20250703/share/texmf-var/web2c/texmf.cnf" 1 44455 00ca67f5a06c9c23b32559f3f48cb4e9 "" @@ -45,8 +45,8 @@ "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty" 1 4474 c510a88aa5f51b8c773b50a7ee92befd "" "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amstext.sty" 1 2444 9983e1d0683f102e3b190c64a49313aa "" "/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def" 1 30351 a2b09edc6c93a742566b222c33d0278e "" - "paper.aux" 1777106347.22354 429 e867892bc6d7fe276fbd19bcc4f1bc53 "pdflatex" - "paper.tex" 1777106346.52452 7702 1354320d93134a233d8c3c035e09f7c0 "" + "paper.aux" 1777106636.0643 429 e867892bc6d7fe276fbd19bcc4f1bc53 "pdflatex" + "paper.tex" 1777106635.61829 7628 2263cb80db1ec8aa25bb60c5652b01c2 "" (generated) "paper.aux" "paper.log" diff --git a/papers/plane_depth_sequencing/paper.log b/papers/plane_depth_sequencing/paper.log index 4074572..9735e14 100644 --- a/papers/plane_depth_sequencing/paper.log +++ b/papers/plane_depth_sequencing/paper.log @@ -1,43 +1,32 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:39 +This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 25 APR 2026 04:43 entering extended mode restricted \write18 enabled. + file:line:error style messages enabled. %&-line parsing enabled. -**paper.tex -(./paper.tex +**/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex +(/home/didericis/Code/math-research/papers/plane_depth_sequencing/paper.tex LaTeX2e <2025-06-01> patch level 1 L3 programming layer <2025-06-09> - 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eKe>zYcb~rf^}{cpfBWgpi~kQ5NR4P4JplletAqpq literal 0 HcmV?d00001 diff --git a/papers/plane_depth_sequencing/paper.tex b/papers/plane_depth_sequencing/paper.tex index 6293420..e1c52a5 100644 --- a/papers/plane_depth_sequencing/paper.tex +++ b/papers/plane_depth_sequencing/paper.tex @@ -109,7 +109,7 @@ Let $G$ be a graph with a plane embedding and outer cycle $C$. For each $d \geq \begin{proof} Let $H = G[V_d]$ with the plane embedding inherited from $G$. It suffices to show every vertex of $H$ lies on the outer face of $H$. -For $d = 0$, we have $V_0 = V(C)$ and $H$ is a subgraph of the cycle $C$, hence a disjoint union of paths, which is outerplanar. +For $d = 0$, we have $V_0 = C$, so $H$ is outerplanar. For $d \geq 1$, let $U$ be the open subset of the plane obtained by removing all vertices and edges of $H$. We show every $v \in V_d$ lies on the boundary of the component $U_{\mathrm{out}}$ of $U$ containing the outer face of $G$.