Refute plane diamond coloring conjecture with 16-vertex counterexample
Adds Theorem 2.6 stating Conjecture 2.5 is false, with proof exhibiting
a 16-vertex maximal planar graph (graph6 O???IAGKEBEQQYHdplW{n) for
which the auxiliary 4-colorability check fails at every root vertex,
verified computationally via Sage's chromatic_number. Includes the
graph as a figure and adds a McKay graph6 reference.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
BIN
Binary file not shown.
|
After Width: | Height: | Size: 37 KiB |
@@ -4,12 +4,16 @@
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent }
|
||||
\citation{appel1977every}
|
||||
\citation{robertson1997four}
|
||||
\citation{mckaygraph6}
|
||||
\bibcite{appel1977every}{1}
|
||||
\bibcite{robertson1997four}{2}
|
||||
\bibcite{mckaygraph6}{3}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{2}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{2}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A maximal planar graph on $16$ vertices with no plane diamond coloring.}}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:counterexample}{{1}{3}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{3}
|
||||
|
||||
@@ -1,5 +1,5 @@
|
||||
# Fdb version 3
|
||||
["pdflatex"] 1778344156 "paper.tex" "paper.pdf" "paper" 1778344156
|
||||
["pdflatex"] 1778344761 "paper.tex" "paper.pdf" "paper" 1778344761
|
||||
"/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 ""
|
||||
@@ -20,6 +20,8 @@
|
||||
"/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/cmti7.tfm" 1136768653 1492 86331993fe614793f5e7e755835c31c5 ""
|
||||
"/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/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 ""
|
||||
"/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 ""
|
||||
@@ -33,7 +35,10 @@
|
||||
"/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/cmti7.pfb" 1248133631 36607 d654cb3f2bc54f57509240071db3bffa ""
|
||||
"/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/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c ""
|
||||
"/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/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 ""
|
||||
@@ -43,13 +48,23 @@
|
||||
"/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/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/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
|
||||
"/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 ""
|
||||
"/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" 1778344156 697 3f22ac2880dcfce06793ac851e5b5d36 "pdflatex"
|
||||
"paper.tex" 1778344142 7304 251fb4f84f05f1ed3480fd521a52c92f ""
|
||||
"counterexample.png" 1778344642 38090 f4e4209c691fb5c14386207f1bb29f68 ""
|
||||
"paper.aux" 1778344761 957 a638007d0f0ce9d55a02d7823fdbddc2 "pdflatex"
|
||||
"paper.tex" 1778344755 9964 4b14437133e6c4b02a0c45d6a8579fa8 ""
|
||||
(generated)
|
||||
"paper.aux"
|
||||
"paper.log"
|
||||
|
||||
@@ -93,6 +93,69 @@ 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/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/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/url/url.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
|
||||
@@ -132,6 +195,25 @@ 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
|
||||
@@ -146,6 +228,15 @@ 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/cmti7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm
|
||||
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm
|
||||
INPUT ./counterexample.png
|
||||
INPUT ./counterexample.png
|
||||
INPUT counterexample.png
|
||||
INPUT ./counterexample.png
|
||||
INPUT ./counterexample.png
|
||||
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
|
||||
@@ -160,4 +251,6 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pf
|
||||
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/cmti7.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/cmtt10.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
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 12:29
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 12:39
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -107,6 +107,34 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
\thm@headsep=\skip56
|
||||
\dth@everypar=\toks26
|
||||
)
|
||||
(/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@=\toks27
|
||||
)
|
||||
(/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
|
||||
)
|
||||
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=\dimen150
|
||||
\Gin@req@width=\dimen151
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/url/url.sty
|
||||
\Urlmuskip=\muskip17
|
||||
Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc.
|
||||
)
|
||||
\c@theorem=\count272
|
||||
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
@@ -117,58 +145,102 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
|
||||
(./paper.aux)
|
||||
\openout1 = `paper.aux'.
|
||||
|
||||
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.
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 57.
|
||||
LaTeX Font Info: ... okay on input line 57.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 57.
|
||||
|
||||
(/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 54.
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 57.
|
||||
|
||||
|
||||
(/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]
|
||||
(./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
1725 strings out of 478268
|
||||
24057 string characters out of 5846347
|
||||
325269 words of memory out of 5000000
|
||||
19810 multiletter control sequences out of 15000+600000
|
||||
476364 words of font info for 55 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,6n,76p,613b,223s 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/c
|
||||
m/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/c
|
||||
mr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr
|
||||
7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.p
|
||||
fb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pf
|
||||
b></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/cmti7.pfb></u
|
||||
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr
|
||||
/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (2 pages, 176410 bytes).
|
||||
PDF statistics:
|
||||
81 PDF objects out of 1000 (max. 8388607)
|
||||
48 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)
|
||||
)
|
||||
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count274
|
||||
\scratchdimen=\dimen152
|
||||
\scratchbox=\box54
|
||||
\nofMPsegments=\count275
|
||||
\nofMParguments=\count276
|
||||
\everyMPshowfont=\toks28
|
||||
\MPscratchCnt=\count277
|
||||
\MPscratchDim=\dimen153
|
||||
\MPnumerator=\count278
|
||||
\makeMPintoPDFobject=\count279
|
||||
\everyMPtoPDFconversion=\toks29
|
||||
) (/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 4
|
||||
85.
|
||||
|
||||
(/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 Liv
|
||||
e
|
||||
))
|
||||
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
LaTeX Font Info: Trying to load font information for OMS+cmtt on input line
|
||||
160.
|
||||
LaTeX Font Info: No file OMScmtt.fd. on input line 160.
|
||||
|
||||
|
||||
LaTeX Font Warning: Font shape `OMS/cmtt/m/n' undefined
|
||||
(Font) using `OMS/cmsy/m/n' instead
|
||||
(Font) for symbol `textbraceleft' on input line 160.
|
||||
|
||||
[2]
|
||||
<counterexample.png, id=24, 338.9463pt x 339.669pt>
|
||||
File: counterexample.png Graphic file (type png)
|
||||
<use counterexample.png>
|
||||
Package pdftex.def Info: counterexample.png used on input line 181.
|
||||
(pdftex.def) Requested size: 161.9989pt x 162.34474pt.
|
||||
[3 <./counterexample.png>] (./paper.aux)
|
||||
|
||||
LaTeX Font Warning: Some font shapes were not available, defaults substituted.
|
||||
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
2684 strings out of 478268
|
||||
38809 string characters out of 5846347
|
||||
343139 words of memory out of 5000000
|
||||
20726 multiletter control sequences out of 15000+600000
|
||||
477036 words of font info for 59 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,8n,76p,847b,344s 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></u
|
||||
sr/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/lo
|
||||
cal/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/te
|
||||
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/tex
|
||||
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti7.pfb></usr/local/texli
|
||||
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive
|
||||
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/
|
||||
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/local/texlive/20
|
||||
22/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (3 pages, 236597 bytes).
|
||||
PDF statistics:
|
||||
97 PDF objects out of 1000 (max. 8388607)
|
||||
57 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
6 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -36,6 +36,9 @@
|
||||
|
||||
\documentclass{amsart}
|
||||
|
||||
\usepackage{graphicx}
|
||||
\usepackage{url}
|
||||
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
@@ -147,6 +150,39 @@ The conjecture below asserts a structural property of $4$-colorings of maximal p
|
||||
Every maximal planar graph $G$ has a plane diamond coloring.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{theorem}
|
||||
Conjecture 2.5 is false.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Let $G$ be the maximal planar graph on $16$ vertices with graph6 string\footnote{We use the standard graph6 encoding of McKay; see \cite{mckaygraph6}.}
|
||||
\[
|
||||
\texttt{O???IAGKEBEQQYHdplW\{n}
|
||||
\]
|
||||
shown in Figure~\ref{fig:counterexample}. Equivalently, $G$ has edge set
|
||||
\begin{align*}
|
||||
&\{0,8\}, \{0,10\}, \{0,14\}, \{1,7\}, \{1,10\}, \{1,12\}, \{2,9\}, \{2,11\}, \{2,13\}, \\
|
||||
&\{3,9\}, \{3,11\}, \{3,15\}, \{4,8\}, \{4,12\}, \{4,14\}, \{4,15\}, \{5,6\}, \{5,13\}, \\
|
||||
&\{5,14\}, \{5,15\}, \{6,11\}, \{6,13\}, \{6,15\}, \{7,10\}, \{7,12\}, \{7,14\}, \{8,10\}, \\
|
||||
&\{8,12\}, \{8,14\}, \{9,11\}, \{9,13\}, \{9,15\}, \{10,12\}, \{10,14\}, \{11,13\}, \\
|
||||
&\{11,15\}, \{12,13\}, \{12,14\}, \{12,15\}, \{13,14\}, \{13,15\}, \{14,15\}.
|
||||
\end{align*}
|
||||
We have $|V(G)| = 16$ and $|E(G)| = 42 = 3 \cdot 16 - 6$, so $G$ is a triangulation.
|
||||
|
||||
By Proposition 2.4, it suffices to verify that for every choice of root $u \in V(G)$ and every pair of distinct colors $c_a, c_b$, no proper $4$-coloring $C$ of $G$ satisfies $C^{-1}(c_a) \subseteq \bigcup_{i \text{ even}} L_i^{(u)}$ and $C^{-1}(c_b) \subseteq \bigcup_{i \text{ odd}} L_i^{(u)}$, where $\{L_i^{(u)}\}$ denotes the distance partition from $u$.
|
||||
|
||||
For a fixed root $u$, the existence of such a $4$-coloring $C$ together with colors $c_a, c_b$ is equivalent to $4$-colorability of the auxiliary graph $H_u$ obtained from $G$ by adjoining two new vertices $\alpha, \beta$, joining $\alpha$ to every vertex in $\bigcup_{i \text{ odd}} L_i^{(u)}$, joining $\beta$ to every vertex in $\bigcup_{i \text{ even}} L_i^{(u)}$, and adding the edge $\{\alpha, \beta\}$. Indeed, in any proper $4$-coloring of $H_u$ the colors of $\alpha$ and $\beta$ are distinct and absent from the odd-parity and even-parity layers of $G$ respectively, yielding the required colors $c_a := C(\alpha)$ and $c_b := C(\beta)$. Conversely, given $C, c_a, c_b$ as in the proposition, setting $C(\alpha) := c_a$ and $C(\beta) := c_b$ extends $C$ to a proper $4$-coloring of $H_u$.
|
||||
|
||||
A direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies that $\chi(H_u) > 4$ for every $u \in V(G)$, so $G$ admits no plane diamond coloring.
|
||||
\end{proof}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.45\textwidth]{counterexample.png}
|
||||
\caption{A maximal planar graph on $16$ vertices with no plane diamond coloring.}
|
||||
\label{fig:counterexample}
|
||||
\end{figure}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{appel1977every}
|
||||
@@ -159,6 +195,11 @@ N.~Robertson, D.~Sanders, P.~Seymour, and R.~Thomas,
|
||||
\emph{The four-colour theorem},
|
||||
Journal of Combinatorial Theory, Series B, vol.~70, no.~1, pp.~2--44, 1997.
|
||||
|
||||
\bibitem{mckaygraph6}
|
||||
B.~D.~McKay,
|
||||
\emph{Description of graph6, sparse6 and digraph6 encodings},
|
||||
\url{https://users.cecs.anu.edu.au/~bdm/data/formats.txt}.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user