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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-(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/amsmath/amsgen.sty + (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 generic functions \@emptytoks=\toks17 \ex@=\dimen150 -)) -(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/amsmath/amsbsy.sty +)) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d Bold Symbols \pmbraise@=\dimen151 -) -(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/amsmath/amsopn.sty +) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsmath/amsopn.sty Package: amsopn 2022/04/08 v2.04 operator names ) \inf@bad=\count275 @@ -88,15 +77,10 @@ LaTeX Info: Redefining \Relbar on input line 970. 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LaTeX Info: Redefining \] on input line 2950. ) -LaTeX Font Info: Trying to load font information for U+msa on input line 397 -. - -(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/amsfonts/umsa.fd +LaTeX Font Info: Trying to load font information for U+msa on input line 397. + (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2013/01/14 v3.01 AMS symbols A -) -(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/amsfonts/amsfonts.sty +) (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 @@ -129,9 +113,7 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' \dth@everypar=\toks27 ) \c@theorem=\count295 - -(/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.2025070 -3-texmfdist/tex/latex/l3backend/l3backend-pdftex.def + (/nix/store/zwvq8i154s539b4w2fqhia83fsfng7ng-texlive-combined-full-2025.20250703-texmfdist/tex/latex/l3backend/l3backend-pdftex.def File: l3backend-pdftex.def 2025-06-09 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count296 ) (./paper.aux) @@ -152,22 +134,15 @@ LaTeX Font Info: ... okay on input line 52. 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PDF statistics: 71 PDF objects out of 1000 (max. 8388607) 42 compressed objects within 1 object stream diff --git a/papers/plane_depth_sequencing/paper.pdf b/papers/plane_depth_sequencing/paper.pdf index a2d1293..3223858 100644 Binary files a/papers/plane_depth_sequencing/paper.pdf and b/papers/plane_depth_sequencing/paper.pdf differ diff --git a/papers/plane_depth_sequencing/paper.synctex.gz b/papers/plane_depth_sequencing/paper.synctex.gz new file mode 100644 index 0000000..a5cadce Binary files /dev/null and b/papers/plane_depth_sequencing/paper.synctex.gz differ 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 $G$.