Add kempe proof fixed
This commit is contained in:
@@ -0,0 +1,19 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Kempe's Proof (Valid Portion)}}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Setup}}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{Every Planar Graph Has a Vertex of Degree at Most 5}}{1}{}\protected@file@percent }
|
||||
\@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 }
|
||||
\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}
|
||||
@@ -0,0 +1,59 @@
|
||||
# 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 ""
|
||||
"/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 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 ""
|
||||
"/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/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 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 ""
|
||||
"/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/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
|
||||
"/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 ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
"paper.pdf"
|
||||
@@ -0,0 +1,164 @@
|
||||
PWD /Users/didericis/Code/math-research/kempe_proof_fixed
|
||||
INPUT /usr/local/texlive/2022/texmf.cnf
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
|
||||
INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
|
||||
INPUT /Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex
|
||||
OUTPUT paper.log
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
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/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/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/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/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/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/l3backend/l3backend-pdftex.def
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
INPUT ./paper.aux
|
||||
INPUT paper.aux
|
||||
INPUT paper.aux
|
||||
OUTPUT paper.aux
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
|
||||
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/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
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
|
||||
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
|
||||
OUTPUT paper.pdf
|
||||
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
|
||||
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
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb
|
||||
@@ -0,0 +1,149 @@
|
||||
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
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex
|
||||
(/Users/didericis/Code/math-research/kempe_proof_fixed/paper.tex
|
||||
LaTeX2e <2021-11-15> patch level 1
|
||||
L3 programming layer <2022-02-24> (/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
|
||||
Document Class: amsart 2020/05/29 v2.20.6
|
||||
\linespacing=\dimen138
|
||||
\normalparindent=\dimen139
|
||||
\normaltopskip=\skip47
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
Package: amsmath 2021/10/15 v2.17l AMS math features
|
||||
\@mathmargin=\skip48
|
||||
|
||||
For additional information on amsmath, use the `?' option.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
Package: amstext 2021/08/26 v2.01 AMS text
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
File: amsgen.sty 1999/11/30 v2.0 generic functions
|
||||
\@emptytoks=\toks16
|
||||
\ex@=\dimen140
|
||||
)) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
\pmbraise@=\dimen141
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
Package: amsopn 2021/08/26 v2.02 operator names
|
||||
)
|
||||
\inf@bad=\count185
|
||||
LaTeX Info: Redefining \frac on input line 234.
|
||||
\uproot@=\count186
|
||||
\leftroot@=\count187
|
||||
LaTeX Info: Redefining \overline on input line 399.
|
||||
\classnum@=\count188
|
||||
\DOTSCASE@=\count189
|
||||
LaTeX Info: Redefining \ldots on input line 496.
|
||||
LaTeX Info: Redefining \dots on input line 499.
|
||||
LaTeX Info: Redefining \cdots on input line 620.
|
||||
\Mathstrutbox@=\box50
|
||||
\strutbox@=\box51
|
||||
\big@size=\dimen142
|
||||
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
|
||||
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
|
||||
\macc@depth=\count190
|
||||
\c@MaxMatrixCols=\count191
|
||||
\dotsspace@=\muskip16
|
||||
\c@parentequation=\count192
|
||||
\dspbrk@lvl=\count193
|
||||
\tag@help=\toks17
|
||||
\row@=\count194
|
||||
\column@=\count195
|
||||
\maxfields@=\count196
|
||||
\andhelp@=\toks18
|
||||
\eqnshift@=\dimen143
|
||||
\alignsep@=\dimen144
|
||||
\tagshift@=\dimen145
|
||||
\tagwidth@=\dimen146
|
||||
\totwidth@=\dimen147
|
||||
\lineht@=\dimen148
|
||||
\@envbody=\toks19
|
||||
\multlinegap=\skip49
|
||||
\multlinetaggap=\skip50
|
||||
\mathdisplay@stack=\toks20
|
||||
LaTeX Info: Redefining \[ on input line 2938.
|
||||
LaTeX Info: Redefining \] on input line 2939.
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 397.
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
) (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
|
||||
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
|
||||
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
|
||||
)
|
||||
\copyins=\insert199
|
||||
\abstractbox=\box52
|
||||
\listisep=\skip51
|
||||
\c@part=\count197
|
||||
\c@section=\count198
|
||||
\c@subsection=\count266
|
||||
\c@subsubsection=\count267
|
||||
\c@paragraph=\count268
|
||||
\c@subparagraph=\count269
|
||||
\c@figure=\count270
|
||||
\c@table=\count271
|
||||
\abovecaptionskip=\skip52
|
||||
\belowcaptionskip=\skip53
|
||||
\captionindent=\dimen149
|
||||
\thm@style=\toks21
|
||||
\thm@bodyfont=\toks22
|
||||
\thm@headfont=\toks23
|
||||
\thm@notefont=\toks24
|
||||
\thm@headpunct=\toks25
|
||||
\thm@preskip=\skip54
|
||||
\thm@postskip=\skip55
|
||||
\thm@headsep=\skip56
|
||||
\dth@everypar=\toks26
|
||||
)
|
||||
\c@theorem=\count272
|
||||
(/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)
|
||||
\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.
|
||||
(/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.
|
||||
(/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) )
|
||||
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
|
||||
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).
|
||||
PDF statistics:
|
||||
97 PDF objects out of 1000 (max. 8388607)
|
||||
58 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)
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -0,0 +1,238 @@
|
||||
%% filename: amsart-template.tex
|
||||
%% version: 1.1
|
||||
%% date: 2014/07/24
|
||||
%%
|
||||
%% American Mathematical Society
|
||||
%% Technical Support
|
||||
%% Publications Technical Group
|
||||
%% 201 Charles Street
|
||||
%% Providence, RI 02904
|
||||
%% USA
|
||||
%% tel: (401) 455-4080
|
||||
%% (800) 321-4267 (USA and Canada only)
|
||||
%% fax: (401) 331-3842
|
||||
%% email: tech-support@ams.org
|
||||
%%
|
||||
%% Copyright 2008-2010, 2014 American Mathematical Society.
|
||||
%%
|
||||
%% This work may be distributed and/or modified under the
|
||||
%% conditions of the LaTeX Project Public License, either version 1.3c
|
||||
%% of this license or (at your option) any later version.
|
||||
%% The latest version of this license is in
|
||||
%% http://www.latex-project.org/lppl.txt
|
||||
%% and version 1.3c or later is part of all distributions of LaTeX
|
||||
%% version 2005/12/01 or later.
|
||||
%%
|
||||
%% This work has the LPPL maintenance status `maintained'.
|
||||
%%
|
||||
%% The Current Maintainer of this work is the American Mathematical
|
||||
%% Society.
|
||||
%%
|
||||
%% ====================================================================
|
||||
|
||||
% AMS-LaTeX v.2 template for use with amsart
|
||||
%
|
||||
% Remove any commented or uncommented macros you do not use.
|
||||
|
||||
\documentclass{amsart}
|
||||
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
\newtheorem{example}[theorem]{Example}
|
||||
\newtheorem{xca}[theorem]{Exercise}
|
||||
|
||||
\theoremstyle{remark}
|
||||
\newtheorem{remark}[theorem]{Remark}
|
||||
|
||||
\numberwithin{equation}{section}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{Humans Suffice: A Novel Proof of the Four Color Theorem}
|
||||
|
||||
% Remove any unused author tags.
|
||||
% author one information
|
||||
\author{Eric Bauerfeld}
|
||||
\address{}
|
||||
\curraddr{}
|
||||
\email{eric@dideric.is}
|
||||
\thanks{Dr. Joe Fields}
|
||||
\thanks{Holepunch}
|
||||
|
||||
% author two information
|
||||
% \author{}
|
||||
% \address{}
|
||||
% \curraddr{}
|
||||
% \email{}
|
||||
% \thanks{}
|
||||
|
||||
\subjclass[2020]{Primary }
|
||||
|
||||
\keywords{Discrete , Four Color Theorem,}
|
||||
|
||||
\date{}
|
||||
|
||||
\dedicatory{Dedicated to all who value more than machines}
|
||||
|
||||
\begin{abstract}
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Kempe's Proof (Valid Portion)}
|
||||
|
||||
\subsection{Setup}
|
||||
|
||||
Kempe's strategy, published in 1879, follows a \emph{minimal counterexample} argument. Suppose, for contradiction, that there exists a planar graph requiring 5 colors. Among all such graphs, let $G$ be one with the fewest vertices. Then every planar graph with fewer vertices than $G$ is 4-colorable, but $G$ itself is not.
|
||||
|
||||
\subsection{Every Planar Graph Has a Vertex of Degree at Most 5}
|
||||
|
||||
\begin{lemma}
|
||||
Every planar graph has at least one vertex of degree $\leq 5$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Let $G$ be a connected planar graph with $V$ vertices, $E$ edges, and $F$ faces. By Euler's formula,
|
||||
\[
|
||||
V - E + F = 2.
|
||||
\]
|
||||
Since every face is bounded by at least 3 edges and each edge borders at most 2 faces, we have $2E \geq 3F$, hence $F \leq \frac{2E}{3}$. Substituting into Euler's formula:
|
||||
\[
|
||||
V - E + \frac{2E}{3} \geq 2 \implies E \leq 3V - 6.
|
||||
\]
|
||||
If every vertex had degree $\geq 6$, then $2E \geq 6V$, so $E \geq 3V$, contradicting $E \leq 3V - 6$. Therefore at least one vertex has degree $\leq 5$.
|
||||
\end{proof}
|
||||
|
||||
Since $G$ is a minimal counterexample, it must contain a vertex $v$ of degree at most 5. Kempe argued by cases on the degree of $v$.
|
||||
|
||||
\subsection{Cases of Degree at Most 3}
|
||||
|
||||
Suppose $v$ has degree $\leq 3$. Remove $v$ from $G$ to obtain the graph $G - v$. Since $G - v$ has fewer vertices than $G$, it is 4-colorable by minimality. Fix such a 4-coloring. Now reinsert $v$: its at most 3 neighbors occupy at most 3 of the 4 colors, so at least one color remains available for $v$. This yields a valid 4-coloring of $G$, a contradiction.
|
||||
|
||||
\subsection{Case of Degree 4}
|
||||
|
||||
Suppose $v$ has degree exactly 4 with neighbors $a$, $b$, $c$, $d$ appearing in cyclic order around $v$ in the planar embedding. Remove $v$ and 4-color $G - v$ by minimality. If the four neighbors do not all receive distinct colors, then at least one color is unused among them, and we may assign that color to $v$, giving a contradiction.
|
||||
|
||||
So assume $a$, $b$, $c$, $d$ receive all four distinct colors; call them $1$, $2$, $3$, $4$ respectively. Define a \emph{Kempe chain} to be a maximal connected subgraph whose vertices are colored with exactly two specified colors.
|
||||
|
||||
Consider the Kempe chain $K_{13}$ containing $a$ (using colors 1 and 3).
|
||||
|
||||
\begin{itemize}
|
||||
\item \textbf{Case 1}: $c$ is not in $K_{13}$. Swap colors 1 and 3 throughout $K_{13}$. This is still a valid coloring of $G - v$, and now $a$ receives color 3, so color 1 is free for $v$.
|
||||
|
||||
\item \textbf{Case 2}: $c$ is in $K_{13}$. Then there is a path of alternating colors 1 and 3 from $a$ to $c$ in the planar embedding. Because $a$ and $c$ alternate around $v$ with $b$ and $d$, this path separates $b$ from $d$ in the plane. Therefore $b$ and $d$ lie in different Kempe chains for colors 2 and 4. Swap colors 2 and 4 in the chain containing $b$; now $b$ receives color 4, freeing color 2 for $v$.
|
||||
\end{itemize}
|
||||
|
||||
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}
|
||||
|
||||
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.
|
||||
|
||||
\begin{lemma}
|
||||
If $G$ is a planar graph with minimum degree 5, then $G$ contains at least 12 vertices of degree exactly 5.
|
||||
\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 $n_5 \geq 12$.
|
||||
\end{proof}
|
||||
|
||||
\subsection{Two Non-Adjacent Vertices of Degree 5}
|
||||
|
||||
\begin{lemma}
|
||||
If $G$ is a planar graph with minimum degree 5, then $G$ contains two non-adjacent vertices of degree exactly 5.
|
||||
\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.
|
||||
\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 $G'$.
|
||||
|
||||
\subsection{Not All Colorings in $\Phi$ Saturate Both Neighborhoods}
|
||||
|
||||
\begin{lemma}
|
||||
It is not possible that every $\phi \in \Phi$ satisfies both $|\phi(N(v_0))| = 4$ and $|\phi(N(v_1))| = 4$.
|
||||
\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.
|
||||
\end{proof}
|
||||
|
||||
\section{Merged Subgraphs of $G'$}
|
||||
|
||||
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
|
||||
\[
|
||||
\mathcal{M} = \bigl\{\, G'_{ij} : a_i, a_j \in N(v_0),\ \{a_i, a_j\} \notin E(G') \,\bigr\}.
|
||||
\]
|
||||
|
||||
\subsection{Colorings of Merged Subgraphs Extend to $G'$}
|
||||
|
||||
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}
|
||||
|
||||
\end{document}
|
||||
|
||||
%-----------------------------------------------------------------------
|
||||
% End of amsart-template.tex
|
||||
%-----------------------------------------------------------------------
|
||||
Reference in New Issue
Block a user