From 76b8619bd32ee217241d8be7a312643c60729bbf Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 15 Apr 2026 19:15:03 -0400 Subject: [PATCH] chore: SAVEPOINT --- paper.aux | 7 ++++--- paper.fdb_latexmk | 9 +++++---- paper.fls | 1 + paper.log | 16 ++++++++-------- paper.pdf | Bin 213600 -> 227402 bytes paper.synctex.gz | Bin 43824 -> 52374 bytes paper.tex | 38 ++++++++++++++++++++++++++++++++++++++ 7 files changed, 56 insertions(+), 15 deletions(-) diff --git a/paper.aux b/paper.aux index 36a755f..7cd7ec2 100644 --- a/paper.aux +++ b/paper.aux @@ -10,11 +10,12 @@ \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{The Reduced Subgraph $G'$}}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{Not All Colorings in $\Phi $ Saturate Both Neighborhoods}}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Merged Subgraphs of $G'$}}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Colorings of Merged Subgraphs Extend to $G'$}}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{4-Saturating Merged Colorings Extend to 4-Saturating Colorings of $G'$}}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{Locked Colorings}}{4}{}\protected@file@percent } \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{29.38873pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Colorings of Merged Subgraphs Extend to $G'$}}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{4-Saturating Merged Colorings Extend to 4-Saturating Colorings of $G'$}}{4}{}\protected@file@percent } -\gdef \@abspage@last{4} +\gdef \@abspage@last{5} diff --git a/paper.fdb_latexmk b/paper.fdb_latexmk index 61d2d8c..a21040f 100644 --- a/paper.fdb_latexmk +++ b/paper.fdb_latexmk @@ -1,6 +1,6 @@ # Fdb version 3 -["pdflatex"] 1776052801 "/Users/didericis/Code/magnum-opus-4ct/paper.tex" "paper.pdf" "paper" 1776052801 - "/Users/didericis/Code/magnum-opus-4ct/paper.tex" 1776052800 13483 66b66eccf22ca3f14d4a88ccbcab3f5d "" +["pdflatex"] 1776095794 "/Users/didericis/Code/magnum-opus-4ct/paper.tex" "paper.pdf" "paper" 1776095795 + "/Users/didericis/Code/magnum-opus-4ct/paper.tex" 1776095794 15857 69e25517a47258b6dfdf516081e7ad38 "" "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -25,6 +25,7 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb" 1248133631 31809 8670ca339bf94e56da1fc21c80635e2a "" @@ -52,8 +53,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1776052801 1899 6f6c9f990e281a7651f9bbdc326c1739 "pdflatex" - "paper.tex" 1776052800 13483 66b66eccf22ca3f14d4a88ccbcab3f5d "" + "paper.aux" 1776095795 2015 90a6de9b8264496ca14fb7c71164cfd9 "pdflatex" + "paper.tex" 1776095794 15857 69e25517a47258b6dfdf516081e7ad38 "" (generated) "paper.aux" "paper.log" diff --git a/paper.fls b/paper.fls index d487f18..d0bdfdb 100644 --- a/paper.fls +++ b/paper.fls @@ -150,6 +150,7 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.p INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb diff --git a/paper.log b/paper.log index a8233a2..134bae3 100644 --- a/paper.log +++ b/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) 13 APR 2026 00:00 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 13 APR 2026 11:56 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -140,20 +140,20 @@ Overfull \hbox (18.04422pt too wide) detected at line 257 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 )) = [](\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 a[]; a[]\OMS/cmsy/m/n/10 g [ \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 n f\OML/cmm/m/it/10 a[]; a[]\OMS/cmsy/m/n/10 g\OT1/cmr/m/n/10 )) = \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a[]\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g [ \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 n f\OML/cmm/m/it/10 a[]; a[]\OMS/cmsy/m/n/10 g\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v[]\OT1/cmr/m/n/10 ))\OML/cmm/m/it/10 : [] -[4] (./paper.aux) ) +[4] [5] (./paper.aux) ) Here is how much of TeX's memory you used: 1719 strings out of 478268 24065 string characters out of 5846347 - 330202 words of memory out of 5000000 + 331202 words of memory out of 5000000 19803 multiletter control sequences out of 15000+600000 475834 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,9n,76p,727b,280s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (4 pages, 213600 bytes). + 69i,9n,76p,803b,280s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on paper.pdf (5 pages, 227402 bytes). 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By definition of t The first equality expands $N(v_0)$; the second uses $\widetilde{\psi}(a_i) = \widetilde{\psi}(a_j) = \psi(a_{ij})$ and $\widetilde{\psi}(u) = \psi(u)$ for $u \notin \{a_i, a_j\}$; the third uses $N_{ij}(v_0) = \{a_{ij}\} \cup (N(v_0) \setminus \{a_i,a_j\})$. Therefore $|\widetilde{\psi}(N(v_0))| = |\psi(N_{ij}(v_0))| = 4$. Since also $\widetilde{\psi} \in \Phi$ by the previous lemma, we conclude $\widetilde{\psi} \in \Phi_4$. \end{proof} +\subsection{Locked Colorings} + +\begin{definition} +Let $a_i, a_j \in N(v_0)$ be non-adjacent in $G'$ (where $G' = G - \{v_0, v_1\}$ as in Section 2.1). A coloring $\phi \in \Phi$ is \emph{locked relative to $\{a_i, a_j\}$} if +\[ + |\phi(N_{G'}(a_i))| > 2 \quad \text{or} \quad |\phi(N_{G'}(a_j))| > 2. +\] +Denote the set of all such colorings by $\Lambda_{ij} \subseteq \Phi$. +\end{definition} + +Intuitively, a coloring is locked relative to $\{a_i, a_j\}$ when the neighborhood of at least one of the two vertices is colored with enough distinct colors to obstruct a Kempe chain swap that would free a color for $v_0$. + +\begin{lemma} +Let $G'_{ij} \in \mathcal{M}$ and let $\psi \in \Phi(G'_{ij})$. If $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, then $\psi$ is the induced coloring of some locked coloring in $\Phi$. +\end{lemma} + +\begin{proof} +The extension $\widetilde{\psi} \in \Phi$ (by the lemma in Section 3.1). Suppose for contradiction that $\widetilde{\psi} \notin \Lambda_{ij}$, i.e., $\widetilde{\psi}$ is not locked relative to $\{a_i, a_j\}$. Then +\[ + |\widetilde{\psi}(N_{G'}(a_i))| \leq 2 \quad \text{and} \quad |\widetilde{\psi}(N_{G'}(a_j))| \leq 2. +\] +Since $\widetilde{\psi}$ agrees with $\psi$ on all vertices of $G'_{ij}$, the neighborhoods of $a_i$ and $a_j$ in $G'$ each use at most 2 colors under $\widetilde{\psi}$. With so few colors in each neighborhood, a Kempe chain swap can be performed in $\widetilde{\psi}$ to give $a_i$ and $a_j$ the same color, freeing a fourth color for $v_0$. Simultaneously, since $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, the vertex $v_1$ can be assigned the remaining color not used by its neighbors. Together these assignments extend $\widetilde{\psi}$ to a proper 4-coloring of all of $G$, contradicting the minimality of $G$ as a counterexample. Therefore $\widetilde{\psi} \in \Lambda_{ij}$, and $\psi$ is the induced coloring of the locked coloring $\widetilde{\psi}$. +\end{proof} + +\begin{lemma} + Now prove that all colorings of every merged graph relative to $\{a_i, a_j\}$ must be a locked coloring. +\end{lemma} + +\begin{proof} +\end{proof} + +\begin{lemma} +If all $\phi \in \Phi$ are locked colorings with respect to all pairs of non adjacent vertices $\{a_i, a_j\} \in N(v_0)$, then all colorings of all merged graphs with respect to ${a_k, a_l} \in N(v_1)$ require 4 colors for $N_(v_0)$. +\end{lemma} + +\begin{proof} +\end{proof} + \end{document} %-----------------------------------------------------------------------