didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

chore: SAVEPOINT

Browse Source
This commit is contained in:
didericis
2026-04-17 21:46:57 -04:00
parent a33020ab67
commit 4728c0b62a
7 changed files with 899 additions and 126 deletions
Download Patch File Download Diff File Expand all files Collapse all files
kempe_proof_fixed/paper.aux
+6 -7
View File
@@ -5,15 +5,14 @@
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{Cases of Degree at Most 3}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.4}{Case of Degree 4}}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Resolution of Degree 5 Case}}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{At Least 12 Vertices of Degree 5}}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{Two Non-Adjacent Vertices of Degree 5}}{3}{}\protected@file@percent }
\@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{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The neighborhood of a degree-5 vertex $v_0$: a 5-cycle on $a_0,a_1,a_2,a_3,v_1$ with $v_0$ adjacent to each.}}{2}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The neighborhood of $v_0$ and $v_1$ in $G^*$: $G$ with the edge $\{v_0, v_1\}$ deleted, so the spoke to $v_1$ is absent.}}{3}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The neighborhood of $v_0$ and $v_1$ in $G'$: $G^*$ with $a_0$ and $a_3$ merged into $a_{03}$.}}{3}{}\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 }
\gdef \@abspage@last{4}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces A proper 4-coloring of $G'$ with $\phi (v_0)=\phi (v_1)=\text {red}$, $\phi (a_{03})=\text {blue}$, $\phi (a_1)=\text {green}$, $\phi (a_2)=\text {yellow}$. By the previous lemma, $v_0$ and $v_1$ must lie in both the $\{\text {red},\text {green}\}$-Kempe chain (dashed green, passing through $a_1$) and the $\{\text {red},\text {yellow}\}$-Kempe chain (dashed yellow, passing through $a_2$).}}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The same coloring lifted to $G^*$: $\phi (v_0)=\phi (v_1)=\text {red}$, $\phi (a_0)=\phi (a_3)=\text {blue}$, $\phi (a_1)=\text {green}$, $\phi (a_2)=\text {yellow}$. The $\{\text {red},\text {green}\}$- and $\{\text {red},\text {yellow}\}$-Kempe chains connecting $v_0$ to $v_1$ pass through $a_1$ and $a_2$ respectively.}}{4}{}\protected@file@percent }
\gdef \@abspage@last{5}
kempe_proof_fixed/paper.fdb_latexmk
+84 -7
View File
@@ -1,6 +1,6 @@
# Fdb version 3
["pdflatex"] 1776401949 "/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex" "paper.pdf" "paper" 1776401949
"/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex" 1776401948 11909 77ad6735b0f8e9e4b012ee4a01fe5b9b ""
["pdflatex"] 1776469912 "/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex" "paper.pdf" "paper" 1776469912
"/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex" 1776469911 17355 0d72a35e33a0ef8170e2ce10848261b5 ""
"/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 ""
@@ -23,20 +23,71 @@
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm" 1136768653 768 d7b9a2629a0c353102ad947dc9221d49 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 ""
"/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/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 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb" 1248133631 32915 7bf7720c61a5b3a7ff25b0964421c9b6 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb" 1248133631 24287 6b803fa9eb1ddff9112e00519b09dd9e ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 ""
"/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1601326656 992 855ff26741653ab54814101ca36e153c ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1601326656 19324 f4e4c6403dd0f1605fd20ed22fa79dea ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1601326656 6038 ccb406740cc3f03bbfb58ad504fe8c27 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1601326656 6944 e12f8f7a7364ddf66f93ba30fb3a3742 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1601326656 4883 42daaf41e27c3735286e23e48d2d7af9 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1601326656 2544 8c06d2a7f0f469616ac9e13db6d2f842 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1601326656 44195 5e390c414de027626ca5e2df888fa68d ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1601326656 17311 2ef6b2e29e2fc6a2fc8d6d652176e257 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1601326656 21302 788a79944eb22192a4929e46963a3067 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1601326656 9690 01feb7cde25d4293ef36eef45123eb80 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1601326656 33335 dd1fa4814d4e51f18be97d88bf0da60c ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1601326656 2965 4c2b1f4e0826925746439038172e5d6f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1601326656 5196 2cc249e0ee7e03da5f5f6589257b1e5b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1601326656 20726 d4c8db1e2e53b72721d29916314a22ea ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1601326656 35249 abd4adf948f960299a4b3d27c5dddf46 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1601326656 21989 fdc867d05d228316de137a9fc5ec3bbe ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1601326656 8893 e851de2175338fdf7c17f3e091d94618 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1608933718 11518 738408f795261b70ce8dd47459171309 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1621110968 186007 6e7dfe0bd57520fd5f91641aa72dcac8 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1601326656 32995 ac577023e12c0e4bd8aa420b2e852d1a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1601326656 521 8e224a7af69b7fee4451d1bf76b46654 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1601326656 13391 84d29568c13bdce4133ab4a214711112 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1601326656 104935 184ed87524e76d4957860df4ce0cd1c3 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1601326656 10165 cec5fa73d49da442e56efc2d605ef154 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1601326656 28178 41c17713108e0795aac6fef3d275fbca ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1601326656 9989 c55967bf45126ff9b061fa2ca0c4694f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1601326656 3865 ac538ab80c5cf82b345016e474786549 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1621110968 11024 0179538121bc2dba172013a3ef89519f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1608933718 7854 4176998eeefd8745ac6d2d4bd9c98451 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1601326656 3379 781797a101f647bab82741a99944a229 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1601326656 92405 f515f31275db273f97b9d8f52e1b0736 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1601326656 37376 11cd75aac3da1c1b152b2848f30adc14 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1601326656 8471 c2883569d03f69e8e1cabfef4999cfd7 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1601326656 21201 08d231a2386e2b61d64641c50dc15abd ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1601326656 16121 346f9013d34804439f7436ff6786cef7 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1621110968 44784 cedaa399d15f95e68e22906e2cc09ef8 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1621110968 465 d68603f8b820ea4a08cce534944db581 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1601326656 926 2963ea0dcf6cc6c0a770b69ec46a477b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1601326656 5546 f3f24d7898386cb7daac70bdd2c4d6dc ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1601326656 12601 4786e597516eddd82097506db7cfa098 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1621110968 61163 9b2eefc24e021323e0fc140e9826d016 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1601326656 1896 b8e0ca0ac371d74c0ca05583f6313c91 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1601326656 7778 53c8b5623d80238f6a20aa1df1868e63 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1606168878 23997 a4bed72405fa644418bea7eac2887006 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1621110968 37060 797782f0eb50075c9bc952374d9a659a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1601326656 37431 9abe862035de1b29c7a677f3205e3d9f ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1601326656 4494 af17fb7efeafe423710479858e42fa7e ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1601326656 7251 fb18c67117e09c64de82267e12cd8aa4 ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1621110968 29274 e15c5b7157d21523bd9c9f1dfa146b8e ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1621110968 6825 a2b0ea5b539dda0625e99dd15785ab59 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff ""
@@ -46,13 +97,39 @@
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty" 1636758526 4128 8eea906621b6639f7ba476a472036bbe ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty" 1636758526 2444 926f379cc60fcf0c6e3fee2223b4370d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def" 1601931164 19103 48d29b6e2a64cb717117ef65f107b404 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty" 1622581934 18399 7e40f80366dffb22c0e7b70517db5cb4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty" 1636758526 7996 a8fb260d598dcaf305a7ae7b9c3e3229 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty" 1622581934 2671 4de6781a30211fe0ea4c672e4a2a8166 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty" 1636758526 4009 187ea2dc3194cd5a76cd99a8d7a6c4d0 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex" 1496704379 31293 9b70a8980e3820f030348ac23ff1b1e4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pgflibraryhobby.code.tex" 1456011456 9569 37c145e428b1a33762170e5b67deb774 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty" 1456011456 11598 4fe1d8868a2af2ae05aafb760ef2aaf2 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/tikzlibraryhobby.code.tex" 1496704379 20308 0c8574c87107fd46dbca1bbd8a1fcfb5 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty" 1645736062 6107 e082261f9677386795d8bf6b6254fa9d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty" 1642022539 6758 7d9d899cbbfc962fbc4bb93f4c69eec2 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1601326656 1090 bae35ef70b3168089ef166db3e66f5b2 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1601326656 410 615550c46f918fcbee37641b02a862d9 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1601326656 21013 f4ff83d25bb56552493b030f27c075ae ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1601326656 989 c49c8ae06d96f8b15869da7428047b1e ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1601326656 339 c2e180022e3afdb99c7d0ea5ce469b7d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1601326656 306 c56a323ca5bf9242f54474ced10fca71 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1601326656 443 8c872229db56122037e86bcda49e14f3 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1601326656 348 ee405e64380c11319f0e249fed57e6c5 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1601326656 274 5ae372b7df79135d240456a1c6f2cf9a ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1601326656 325 f9f16d12354225b7dd52a3321f085955 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty" 1635798903 56029 3f7889dab51d620aa43177c391b7b190 ""
"/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 ""
"/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" 1776401949 1729 4a5d59eb7625a23edd5873cf3e508f6e "pdflatex"
"paper.tex" 1776401948 11909 77ad6735b0f8e9e4b012ee4a01fe5b9b ""
"paper.aux" 1776469912 2484 6b86a1f9a86fe81aeebb5158e9ed3df3 "pdflatex"
"paper.tex" 1776469911 17355 0d72a35e33a0ef8170e2ce10848261b5 ""
(generated)
"paper.aux"
"paper.log"
kempe_proof_fixed/paper.fls
+333 -3
View File
@@ -93,6 +93,296 @@ INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/tikzlibraryhobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/tikzlibraryhobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pgflibraryhobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pgflibraryhobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
@@ -104,6 +394,30 @@ INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.de
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
INPUT ./paper.aux
INPUT paper.aux
INPUT paper.aux
@@ -132,6 +446,25 @@ INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm
@@ -148,15 +481,12 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm
INPUT paper.aux
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
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/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
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb
kempe_proof_fixed/paper.log
+291 -32
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 APR 2026 00:59
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 APR 2026 19:51
entering extended mode
restricted \write18 enabled.
file:line:error style messages enabled.
@@ -100,50 +100,309 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
\thm@postskip=\skip55
\thm@headsep=\skip56
\dth@everypar=\toks26
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex
\pgfutil@everybye=\toks27
\pgfutil@tempdima=\dimen150
\pgfutil@tempdimb=\dimen151
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def
\pgfutil@abb=\box53
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex)
Package: pgfrcs 2021/05/15 v3.1.9a (3.1.9a)
))
Package: pgf 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
\KV@toks@=\toks28
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
)
\c@theorem=\count272
Package graphics Info: Driver file: pdftex.def on input line 107.
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
))
\Gin@req@height=\dimen152
\Gin@req@width=\dimen153
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex
Package: pgfsys 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
\pgfkeys@pathtoks=\toks29
\pgfkeys@temptoks=\toks30
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex
\pgfkeys@tmptoks=\toks31
))
\pgf@x=\dimen154
\pgf@y=\dimen155
\pgf@xa=\dimen156
\pgf@ya=\dimen157
\pgf@xb=\dimen158
\pgf@yb=\dimen159
\pgf@xc=\dimen160
\pgf@yc=\dimen161
\pgf@xd=\dimen162
\pgf@yd=\dimen163
\w@pgf@writea=\write3
\r@pgf@reada=\read2
\c@pgf@counta=\count272
\c@pgf@countb=\count273
\c@pgf@countc=\count274
\c@pgf@countd=\count275
\t@pgf@toka=\toks32
\t@pgf@tokb=\toks33
\t@pgf@tokc=\toks34
\pgf@sys@id@count=\count276
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg
File: pgf.cfg 2021/05/15 v3.1.9a (3.1.9a)
)
Driver file for pgf: pgfsys-pdftex.def
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def
File: pgfsys-pdftex.def 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def
File: pgfsys-common-pdf.def 2021/05/15 v3.1.9a (3.1.9a)
))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex
File: pgfsyssoftpath.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfsyssoftpath@smallbuffer@items=\count277
\pgfsyssoftpath@bigbuffer@items=\count278
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex
File: pgfsysprotocol.code.tex 2021/05/15 v3.1.9a (3.1.9a)
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty
Package: xcolor 2021/10/31 v2.13 LaTeX color extensions (UK)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg
File: color.cfg 2016/01/02 v1.6 sample color configuration
)
Package xcolor Info: Driver file: pdftex.def on input line 227.
Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1352.
Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1356.
Package xcolor Info: Model `RGB' extended on input line 1368.
Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1370.
Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1371.
Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1372.
Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1373.
Package xcolor Info: Model `Gray' substituted by `gray' on input line 1374.
Package xcolor Info: Model `wave' substituted by `hsb' on input line 1375.
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex
Package: pgfcore 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex
\pgfmath@dimen=\dimen164
\pgfmath@count=\count279
\pgfmath@box=\box54
\pgfmath@toks=\toks35
\pgfmath@stack@operand=\toks36
\pgfmath@stack@operation=\toks37
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex
\c@pgfmathroundto@lastzeros=\count280
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex
File: pgfcorepoints.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@picminx=\dimen165
\pgf@picmaxx=\dimen166
\pgf@picminy=\dimen167
\pgf@picmaxy=\dimen168
\pgf@pathminx=\dimen169
\pgf@pathmaxx=\dimen170
\pgf@pathminy=\dimen171
\pgf@pathmaxy=\dimen172
\pgf@xx=\dimen173
\pgf@xy=\dimen174
\pgf@yx=\dimen175
\pgf@yy=\dimen176
\pgf@zx=\dimen177
\pgf@zy=\dimen178
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex
File: pgfcorepathconstruct.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@path@lastx=\dimen179
\pgf@path@lasty=\dimen180
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex
File: pgfcorepathusage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@shorten@end@additional=\dimen181
\pgf@shorten@start@additional=\dimen182
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex
File: pgfcorescopes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfpic=\box55
\pgf@hbox=\box56
\pgf@layerbox@main=\box57
\pgf@picture@serial@count=\count281
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex
File: pgfcoregraphicstate.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgflinewidth=\dimen183
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex
File: pgfcoretransformations.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@pt@x=\dimen184
\pgf@pt@y=\dimen185
\pgf@pt@temp=\dimen186
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex
File: pgfcorequick.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex
File: pgfcoreobjects.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex
File: pgfcorepathprocessing.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex
File: pgfcorearrows.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfarrowsep=\dimen187
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex
File: pgfcoreshade.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@max=\dimen188
\pgf@sys@shading@range@num=\count282
\pgf@shadingcount=\count283
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex
File: pgfcoreimage.code.tex 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex
File: pgfcoreexternal.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfexternal@startupbox=\box58
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex
File: pgfcorelayers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex
File: pgfcoretransparency.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex
File: pgfcorepatterns.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex
File: pgfcorerdf.code.tex 2021/05/15 v3.1.9a (3.1.9a)
))) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex
File: pgfmoduleshapes.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfnodeparttextbox=\box59
) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex
File: pgfmoduleplot.code.tex 2021/05/15 v3.1.9a (3.1.9a)
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty
Package: pgfcomp-version-0-65 2021/05/15 v3.1.9a (3.1.9a)
\pgf@nodesepstart=\dimen189
\pgf@nodesepend=\dimen190
) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty
Package: pgfcomp-version-1-18 2021/05/15 v3.1.9a (3.1.9a)
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex)) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex
Package: pgffor 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)
\pgffor@iter=\dimen191
\pgffor@skip=\dimen192
\pgffor@stack=\toks38
\pgffor@toks=\toks39
)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex
Package: tikz 2021/05/15 v3.1.9a (3.1.9a)
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex
File: pgflibraryplothandlers.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgf@plot@mark@count=\count284
\pgfplotmarksize=\dimen193
)
\tikz@lastx=\dimen194
\tikz@lasty=\dimen195
\tikz@lastxsaved=\dimen196
\tikz@lastysaved=\dimen197
\tikz@lastmovetox=\dimen198
\tikz@lastmovetoy=\dimen256
\tikzleveldistance=\dimen257
\tikzsiblingdistance=\dimen258
\tikz@figbox=\box60
\tikz@figbox@bg=\box61
\tikz@tempbox=\box62
\tikz@tempbox@bg=\box63
\tikztreelevel=\count285
\tikznumberofchildren=\count286
\tikznumberofcurrentchild=\count287
\tikz@fig@count=\count288
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex
File: pgfmodulematrix.code.tex 2021/05/15 v3.1.9a (3.1.9a)
\pgfmatrixcurrentrow=\count289
\pgfmatrixcurrentcolumn=\count290
\pgf@matrix@numberofcolumns=\count291
)
\tikz@expandcount=\count292
(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex
File: tikzlibrarytopaths.code.tex 2021/05/15 v3.1.9a (3.1.9a)
))) (/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/tikzlibraryhobby.code.tex (/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pgflibraryhobby.code.tex (/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/hobby.code.tex (/usr/local/texlive/2022/texmf-dist/tex/latex/l3kernel/expl3.sty
Package: expl3 2022-02-24 L3 programming layer (loader)
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
\l__color_backend_stack_int=\count273
\l__pdf_internal_box=\box53
) (./paper.aux)
\l__color_backend_stack_int=\count293
\l__pdf_internal_box=\box64
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/l3packages/xparse/xparse.sty
Package: xparse 2022-01-12 L3 Experimental document command parser
) (/usr/local/texlive/2022/texmf-dist/tex/latex/hobby/pml3array.sty
\l_array_base_int=\count294
\l_array_top_int=\count295
\l_array_tmp_int=\count296
\g_array_base_int=\count297
)
\l_hobby_npoints_int=\count298
\l_hobby_draw_int=\count299
)))
\c@theorem=\count300
(./paper.aux)
\openout1 = `paper.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 52.
LaTeX Font Info: ... okay on input line 52.
LaTeX Font Info: Trying to load font information for U+msa on input line 52.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 54.
LaTeX Font Info: ... okay on input line 54.
LaTeX Font Info: Trying to load font information for U+msa on input line 54.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
LaTeX Font Info: Trying to load font information for U+msb on input line 52.
LaTeX Font Info: Trying to load font information for U+msb on input line 54.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] (./paper.aux) )
) (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count301
\scratchdimen=\dimen259
\scratchbox=\box65
\nofMPsegments=\count302
\nofMParguments=\count303
\everyMPshowfont=\toks40
\MPscratchCnt=\count304
\MPscratchDim=\dimen260
\MPnumerator=\count305
\makeMPintoPDFobject=\count306
\everyMPtoPDFconversion=\toks41
) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485.
(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live
)) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
LaTeX Warning: `h' float specifier changed to `ht'.
LaTeX Warning: `h' float specifier changed to `ht'.
[2]
LaTeX Warning: `h' float specifier changed to `ht'.
LaTeX Warning: `h' float specifier changed to `ht'.
[3]
LaTeX Warning: `h' float specifier changed to `ht'.
[4] [5] (./paper.aux) )
Here is how much of TeX's memory you used:
1719 strings out of 478268
24113 string characters out of 5846347
330202 words of memory out of 5000000
19803 multiletter control sequences out of 15000+600000
13916 strings out of 478268
284468 string characters out of 5846347
572433 words of memory out of 5000000
31734 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,743b,280s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (4 pages, 208777 bytes).
100i,8n,113p,795b,704s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (5 pages, 179482 bytes).
PDF statistics:
97 PDF objects out of 1000 (max. 8388607)
58 compressed objects within 1 object stream
88 PDF objects out of 1000 (max. 8388607)
54 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
1 words of extra memory for PDF output out of 10000 (max. 10000000)
13 words of extra memory for PDF output out of 10000 (max. 10000000)
kempe_proof_fixed/paper.pdf
BIN
View File
Binary file not shown.
kempe_proof_fixed/paper.synctex.gz
BIN
View File
Binary file not shown.
kempe_proof_fixed/paper.tex
+185 -77
View File
@@ -35,6 +35,8 @@
% Remove any commented or uncommented macros you do not use.
\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{hobby}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
@@ -75,7 +77,7 @@
\date{}
\dedicatory{Dedicated to all who value more than machines}
\dedicatory{Dedicated to those who doubt the machine}
\begin{abstract}
\end{abstract}
@@ -130,106 +132,212 @@ In both cases we obtain a valid 4-coloring of $G$, a contradiction.
\section{Resolution of Degree 5 Case}
\subsection{At Least 12 Vertices of Degree 5}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_0$}] (a0) at (90:1.8) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$a_3$}] (a3) at (306:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw (v0) -- (v1);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{The neighborhood of a degree-5 vertex $v_0$: a 5-cycle on $a_0,a_1,a_2,a_3,v_1$ with $v_0$ adjacent to each.}
\end{figure}
Since we have already handled every vertex of degree $\leq 4$, we may assume without loss of generality that the minimal counterexample $G$ has minimum degree 5. We now show that $G$ must contain at least 12 vertices of degree exactly 5.
Let $G$ be a minimum example of a maximal planar graph requiring 5 colors, and let $v_0$ be a graph in $G$ of degree 5, and let $v_1$ be a neighbor of $v_0$.
\begin{lemma}
If $G$ is a planar graph with minimum degree 5, then $G$ contains at least 12 vertices of degree exactly 5.
Let $G^*$ be the subgraph of $G$ obtained by deleting the edge $\{v_0, v_1\}$, and let $G'$ be the graph obtained by merging vertices $a_0$ and $a_3$ in $G^*$. Then there is at least one proper 4-coloring of $G*$ which assigns $v_0$ and $v_1$ the same color and $a_0$ and $a_3$ the same color.
\end{lemma}
\begin{proof}
For each $k \geq 5$, let $n_k$ denote the number of vertices of degree exactly $k$ in $G$. Since every vertex has degree $\geq 5$,
\[
V = \sum_{k \geq 5} n_k, \qquad 2E = \sum_{k \geq 5} k\, n_k.
\]
Using $E \leq 3V - 6$, we obtain $2E \leq 6V - 12$, so
\[
\sum_{k \geq 5} k\, n_k \leq 6\sum_{k \geq 5} n_k - 12.
\]
Moving all terms to the right-hand side gives
\[
12 \leq 6\sum_{k \geq 5} n_k - \sum_{k \geq 5} k\, n_k = \sum_{k \geq 5} (6 - k)\, n_k.
\]
We now expand this sum by separating the $k = 5$, $k = 6$, and $k \geq 7$ terms:
\[
\sum_{k \geq 5}(6-k)\,n_k
= \underbrace{(6-5)\,n_5}_{=\,n_5}
+ \underbrace{(6-6)\,n_6}_{=\,0}
+ \sum_{k \geq 7}(6-k)\,n_k
= n_5 - \sum_{k \geq 7}(k-6)\,n_k.
\]
Therefore $12 \leq n_5 - \sum_{k \geq 7}(k-6)\,n_k$, which rearranges to
\[
n_5 \geq 12 + \sum_{k \geq 7}(k-6)\,n_k.
\]
Since $k - 6 \geq 1 > 0$ and $n_k \geq 0$ for all $k \geq 7$, each term in the sum is non-negative, so $\sum_{k \geq 7}(k-6)\,n_k \geq 0$ and thus $n_5 \geq 12$.
Since $G'$ has fewer vertices than $G$, it admits a proper 4-coloring $\phi$ by minimality. Any such $\phi$ must assign $v_0$ and $v_1$ the same color: if they received distinct colors, $\phi$ would induce a proper 4-coloring of $G^*$ in which $v_0 \neq v_1$, and re-inserting edge $\{v_0,v_1\}$ would yield a proper 4-coloring of $G$, contradicting minimality. The induced coloring $\Phi$ of $G^*$ assigns $a_0$ and $a_3$ the common color $\phi(a_{03})$ by construction, so $\Phi$ is the required coloring.
\end{proof}
\subsection{Two Non-Adjacent Vertices of Degree 5}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_0$}] (a0) at (90:1.8) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$a_3$}] (a3) at (306:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{The neighborhood of $v_0$ and $v_1$ in $G^*$: $G$ with the edge $\{v_0, v_1\}$ deleted, so the spoke to $v_1$ is absent.}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_{03}$}] (a03) at (18:0.9) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a03) -- (a1) -- (a2) -- (a03);
\draw (a03) -- (v1);
\draw (v0) -- (a03);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw[dotted] (a03) -- ++(75:0.7) (a03) -- ++(105:0.7);
\draw[dotted] (a03) -- ++(255:0.7) (a03) -- ++(285:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{The neighborhood of $v_0$ and $v_1$ in $G'$: $G^*$ with $a_0$ and $a_3$ merged into $a_{03}$.}
\end{figure}
\begin{lemma}
If $G$ is a planar graph with minimum degree 5, then $G$ contains two non-adjacent vertices of degree exactly 5.
Let $\Phi$ be a proper 4-coloring of $G*$ induced by a proper 4-coloring of $G'$, let $\alpha = \Phi(v_0) = \Phi(v_1)$, and let $\beta = \Phi(a_{0}) = \Phi(a_{3})$. Then for each of the two remaining colors $\gamma$ and $\delta$, there exists a Kempe chain in $G'$ connecting $v_0$ to $v_1$ using colors $\{\alpha, \gamma\}$ and $\{\alpha, \delta\}$ respectively.
\end{lemma}
\begin{proof}
By the previous lemma, $G$ contains at least 12 vertices of degree exactly 5. Let $S$ denote the set of all degree-5 vertices, so $|S| \geq 12$. Suppose for contradiction that every two vertices in $S$ are adjacent, i.e., $S$ induces a clique in $G$. Then every vertex $v \in S$ is adjacent to all other $|S| - 1 \geq 11$ vertices of $S$. But $\deg(v) = 5$, so $v$ has at most 5 neighbors in total. Since $11 > 5$, this is a contradiction. Therefore $S$ does not form a clique, and there exist two vertices $v_1, v_2 \in S$ that are non-adjacent.
We show $v_0$ and $v_1$ lie in the same $\{\alpha, \gamma\}$-Kempe chain; the argument for $\{\alpha, \delta\}$ is identical.
By the previous lemma, $\Phi$ must exist. Suppose for contradiction that $v_0$ and $v_1$ lie in \emph{different} $\{\alpha, \gamma\}$-Kempe chains. Swapping colors $\alpha$ and $\gamma$ throughout $v_0$'s chain yields a new proper 4-coloring $\Phi'$ of $G*$ in which $\Phi'(v_0) = \gamma$ while $\Phi'(v_1) = \alpha$. Since $\gamma \neq \alpha$, the coloring $\Phi'$ assigns $v_0$ and $v_1$ distinct colors, which allows us to properly color $G$. Therefore $v_0$ and $v_1$ lie in the same $\{\alpha, \gamma\}$-Kempe chain, and by the same argument they lie in the same $\{\alpha, \delta\}$-Kempe chain.
\end{proof}
\subsection{The Reduced Subgraph $G'$}
By the previous lemma, we may fix two non-adjacent vertices $v_0, v_1 \in G$ each of degree 5. Define
\[
G' = G - \{v_0, v_1\},
\]
as the subgraph of $G$ obtained by deleting $v_0$ and $v_1$ together with all edges incident to either vertex. Since $G'$ has strictly fewer vertices than $G$, the minimality of $G$ guarantees that $G'$ admits a proper 4-coloring $\phi : V(G') \to \{1,2,3,4\}$. Because $v_0$ and $v_1$ are non-adjacent in $G$, neither is a neighbor of the other, so each retains all 5 of its neighbors in $G'$. Let $N(v_i)$ denote the set of neighbors of $v_i$ in $G$; then $N(v_0)$ and $N(v_1)$ are each sets of 5 vertices, all of which lie in $G'$ and are assigned colors by $\phi$. Note also that $N(v_0)$ and $N(v_1)$ may overlap. Let $\Phi$ be the set of all possible 4-colorings of $G'$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (v0pos) at (0,0);
\coordinate (a03pos) at (18:0.9);
\coordinate (a1pos) at (162:1.8);
\coordinate (a2pos) at (234:1.8);
\coordinate (v1pos) at (18:1.8);
% Kempe chain highlights beneath nodes
\draw[line width=4pt, green!50, dashed] (a1pos) to[bend left=40] (v1pos);
\draw[line width=4pt, green!80, dashed] (a1pos) to (v0pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to[bend right=40] (v1pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to (v0pos);
% Regular edges
\draw (a03pos) -- (a1pos) -- (a2pos) -- (a03pos);
\draw (a03pos) -- (v1pos);
\draw (v0pos) -- (a03pos);
\draw (v0pos) -- (a1pos);
\draw (v0pos) -- (a2pos);
% Dotted lines
\draw[dotted] (a03pos) -- ++(75:0.7) (a03pos) -- ++(105:0.7);
\draw[dotted] (a03pos) -- ++(255:0.7) (a03pos) -- ++(285:0.7);
\draw[dotted] (a1pos) -- ++(147:0.7) (a1pos) -- ++(177:0.7);
\draw[dotted] (a2pos) -- ++(219:0.7) (a2pos) -- ++(249:0.7);
\draw[dotted] (v1pos) -- ++(3:0.7) (v1pos) -- ++(33:0.7);
% Colored nodes
\node[circle, draw, fill=red!60, inner sep=2pt, minimum size=10pt, label=below:{$v_0$}] at (v0pos) {};
\node[circle, draw, fill=blue!60, inner sep=2pt, minimum size=10pt, label=above:{$a_{03}$}] at (a03pos) {};
\node[circle, draw, fill=green!60, inner sep=2pt, minimum size=10pt, label=left:{$a_1$}] at (a1pos) {};
\node[circle, draw, fill=yellow!80, inner sep=2pt, minimum size=10pt, label=left:{$a_2$}] at (a2pos) {};
\node[circle, draw, fill=red!60, inner sep=2pt, minimum size=10pt, label=right:{$v_1$}] at (v1pos) {};
\end{tikzpicture}
\caption{A proper 4-coloring of $G'$ with $\phi(v_0)=\phi(v_1)=\text{red}$, $\phi(a_{03})=\text{blue}$, $\phi(a_1)=\text{green}$, $\phi(a_2)=\text{yellow}$. By the previous lemma, $v_0$ and $v_1$ must lie in both the $\{\text{red},\text{green}\}$-Kempe chain (dashed green, passing through $a_1$) and the $\{\text{red},\text{yellow}\}$-Kempe chain (dashed yellow, passing through $a_2$).}
\end{figure}
\subsection{Not All Colorings in $\Phi$ Saturate Both Neighborhoods}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, inner sep=2pt, minimum size=10pt}]
\coordinate (v0pos) at (0,0);
\coordinate (a0pos) at (90:1.8);
\coordinate (a1pos) at (162:1.8);
\coordinate (a2pos) at (234:1.8);
\coordinate (a3pos) at (306:1.8);
\coordinate (v1pos) at (18:1.8);
\coordinate (green_mid) at (0, 2.4);
\coordinate (yellow_mid) at (306:2.3);
% Kempe chain highlights beneath nodes
\draw[line width=4pt, green!50, dashed] (a1pos) to[curve through={(green_mid)}] (v1pos);
\draw[line width=4pt, green!80, dashed] (a1pos) to (v0pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to[curve through={(yellow_mid)}] (v1pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to (v0pos);
\node[fill=red!60, label=below:{$v_0$}] (v0) at (v0pos) {};
\node[fill=blue!60, label=above:{$a_0$}] (a0) at (a0pos) {};
\node[fill=green!60, label=left:{$a_1$}] (a1) at (a1pos) {};
\node[fill=yellow!80, label=left:{$a_2$}] (a2) at (a2pos) {};
\node[fill=blue!60, label=right:{$a_3$}] (a3) at (a3pos) {};
\node[fill=red!60, label=right:{$v_1$}] (v1) at (v1pos) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{The same coloring lifted to $G^*$: $\phi(v_0)=\phi(v_1)=\text{red}$, $\phi(a_0)=\phi(a_3)=\text{blue}$, $\phi(a_1)=\text{green}$, $\phi(a_2)=\text{yellow}$. The $\{\text{red},\text{green}\}$- and $\{\text{red},\text{yellow}\}$-Kempe chains connecting $v_0$ to $v_1$ pass through $a_1$ and $a_2$ respectively.}
\end{figure}
\begin{lemma}
It is not possible that every $\phi \in \Phi$ satisfies both $|\phi(N(v_0))| = 4$ and $|\phi(N(v_1))| = 4$.
Let $\Phi$ be a proper 4-coloring of $G*$ induced by a proper 4-coloring of $G'$, let $\alpha = \Phi(v_0) = \Phi(v_1)$, let $\beta = \Phi(a_{0}) = \Phi(a_{3})$, let $\delta = \Phi(a_{2})$, and let $\gamma = \Phi(a_{1})$. Then the coloring $\Phi'$ obtained by doing a Kempe switch on the $\{\beta, \delta\}$-Kempe chain incident to $a_0$ must assign $\alpha = \Phi'(v_0) = \Phi'(v_1)$, $\beta = \Phi'(a_3)$, $\delta = \Phi'(a_0) = \Phi'(a_2)$, and $\gamma = \Phi'(a_1)$ (IE the coloring for all vertices except $a_0$ remains the same)
\end{lemma}
\begin{proof}
We show the stronger fact that there exists $\phi \in \Phi$ with $|\phi(N(v_0))| \leq 3$.
Since $v_1$ is a single vertex, $G - \{v_1\}$ is a planar graph on $V - 1$ vertices. Because $V - 1 < V$ and $G$ is a minimal counterexample, $G - \{v_1\}$ is 4-colorable. Fix a proper 4-coloring $\psi$ of $G - \{v_1\}$. This coloring assigns a color to every vertex of $G'$ and also to $v_0$.
Let $\phi = \psi|_{V(G')}$ be the restriction of $\psi$ to $G'$. Since $\psi$ is a proper coloring of $G - \{v_1\} \supseteq G'$, the restriction $\phi$ is a proper 4-coloring of $G'$, so $\phi \in \Phi$.
Because $\psi$ is a proper coloring of $G - \{v_1\}$ and $v_0 \in V(G - \{v_1\})$, the color $\psi(v_0)$ differs from the color of every neighbor of $v_0$. In particular, $\psi(v_0) \notin \phi(N(v_0))$. Since $\psi(v_0) \in \{1,2,3,4\}$ and $\psi(v_0)$ does not appear in $\phi(N(v_0))$, it follows that $|\phi(N(v_0))| \leq 3$. Therefore not every coloring in $\Phi$ uses all 4 colors on $N(v_0)$, and in particular it is impossible that every $\phi \in \Phi$ saturates both $N(v_0)$ and $N(v_1)$ with all 4 colors.
The kempe switch will assign $\Phi'(a_0) = \delta$ by definition. By the previous lemma, there must be a $\{\alpha, \gamma\}$ kempe chain incident to $v_0$, $v_1$, and $a_1$. Therefore the $\{\beta, \delta\}$ component incident to $a_2$ and $a_3$ cannot be connected to the $\{\beta, \delta\}$ component incident to $a_0$, so the kempe switch we do to obtain $\Phi'$ will not recolor vertices $a_2$ and $a_3$. Since the $\{\beta, \delta\}$ kempe switch also maintains vertices assigned $\alpha$ and $\gamma$ in $\Phi$, $\Phi'$ will not recolor vertices $a_1$, $v_0$, and $v_1$. Therefore $\alpha = \Phi'(v_0) = \Phi'(v_1)$, $\beta = \Phi'(a_3)$, $\delta = \Phi'(a_0) = \Phi'(a_2)$, and $\gamma = \Phi'(a_1)$.
\end{proof}
\section{Merged Subgraphs of $G'$}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, inner sep=2pt, minimum size=10pt}]
\coordinate (v0pos) at (0,0);
\coordinate (a0pos) at (90:1.8);
\coordinate (a1pos) at (162:1.8);
\coordinate (a2pos) at (234:1.8);
\coordinate (a3pos) at (306:1.8);
\coordinate (v1pos) at (18:1.8);
Let $N(v_0) = \{a_0, a_1, a_2, a_3, a_4\}$ be the five neighbors of $v_0$ in $G'$. For each pair of non-adjacent vertices $a_i, a_j \in N(v_0)$ (i.e., $\{a_i, a_j\} \notin E(G')$), define the graph $G'_{ij}$ to be the result of identifying $a_i$ and $a_j$ in $G'$: formally, $G'_{ij}$ is obtained from $G'$ by removing $a_i$ and $a_j$ and adding a new vertex $a_{ij}$ adjacent to every vertex in $N_{G'}(a_i) \cup N_{G'}(a_j)$. Define the set of all such merged subgraphs by
\[
\mathcal{M} = \bigl\{\, G'_{ij} : a_i, a_j \in N(v_0),\ \{a_i, a_j\} \notin E(G') \,\bigr\}.
\]
\coordinate (green_mid) at (0, 2.4);
\coordinate (yellow_mid) at (306:2.3);
% Kempe chain highlights beneath nodes
\draw[line width=4pt, green!50, dashed] (a1pos) to[curve through={(green_mid)}] (v1pos);
\draw[line width=4pt, green!80, dashed] (a1pos) to (v0pos);
% \draw[line width=4pt, yellow!80, dashed] (a2pos) to[curve through={(yellow_mid)}] (v1pos);
% \draw[line width=4pt, yellow!80, dashed] (a2pos) to (v0pos);
\subsection{Colorings of Merged Subgraphs Extend to $G'$}
\node[fill=red!60, label=below:{$v_0$}] (v0) at (v0pos) {};
\node[fill=yellow!60, label=above:{$a_0$}] (a0) at (a0pos) {};
\node[fill=green!60, label=left:{$a_1$}] (a1) at (a1pos) {};
\node[fill=yellow!80, label=left:{$a_2$}] (a2) at (a2pos) {};
\node[fill=blue!60, label=right:{$a_3$}] (a3) at (a3pos) {};
\node[fill=red!60, label=right:{$v_1$}] (v1) at (v1pos) {};
For each $G'_{ij} \in \mathcal{M}$, let $\Phi(G'_{ij})$ denote the set of proper 4-colorings of $G'_{ij}$. Each such coloring can be extended to a coloring of $G'$ by assigning the color of the merged vertex $a_{ij}$ back to both $a_i$ and $a_j$. Formally, for $\psi \in \Phi(G'_{ij})$, define $\widetilde{\psi} : V(G') \to \{1,2,3,4\}$ by
\[
\widetilde{\psi}(v) = \begin{cases} \psi(a_{ij}) & \text{if } v = a_i \text{ or } v = a_j, \\ \psi(v) & \text{otherwise.} \end{cases}
\]
Let $\widetilde{\Phi}_{ij} = \{\,\widetilde{\psi} : \psi \in \Phi(G'_{ij})\,\}$ denote the set of all such extensions.
\begin{lemma}
For each $G'_{ij} \in \mathcal{M}$, we have $\widetilde{\Phi}_{ij} \subseteq \Phi$.
\end{lemma}
\begin{proof}
Let $\psi \in \Phi(G'_{ij})$ and let $\widetilde{\psi}$ be its extension to $G'$. We verify that $\widetilde{\psi}$ is a proper coloring of $G'$ by checking every edge $\{u, w\} \in E(G')$.
\textbf{Case 1: neither $u$ nor $w$ is $a_i$ or $a_j$.} Then $\{u, w\}$ is also an edge of $G'_{ij}$, and $\widetilde{\psi}(u) = \psi(u) \neq \psi(w) = \widetilde{\psi}(w)$.
\textbf{Case 2: $u \in \{a_i, a_j\}$ and $w \notin \{a_i, a_j\}$ (or vice versa).} Then $w \in N_{G'}(a_i) \cup N_{G'}(a_j)$, so $w$ is adjacent to $a_{ij}$ in $G'_{ij}$. Since $\psi$ is proper, $\psi(w) \neq \psi(a_{ij}) = \widetilde{\psi}(u)$.
\textbf{Case 3: $u = a_i$ and $w = a_j$.} This case cannot occur, since $a_i$ and $a_j$ are non-adjacent in $G'$ by the definition of $\mathcal{M}$.
In all cases the endpoints of every edge receive distinct colors, so $\widetilde{\psi} \in \Phi$.
\end{proof}
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\end{figure}
\end{document}