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

Add edge-deletion subgraph 4-colorability for a minimal counterexample

Browse Source 
Defines D(G) as the family of single-edge-deletion spanning subgraphs
of a maximal planar graph G, and shows that when G_0 is a minimum-order
5-chromatic maximal planar graph every member of D(G_0) is 4-colorable,
via a coloring pulled back from the smaller minor G_0/uv.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-14 00:27:11 -04:00
parent bd409585ba
commit f6144b98b5
6 changed files with 53 additions and 15 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/flip_symmetric_maximal_planar_graphs/paper.aux
+4 -1
View File
@@ -8,10 +8,13 @@
\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.1}{2}} \newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.1}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Flip symmetry frequency}}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Flip symmetry frequency}}{2}{}\protected@file@percent }
\newlabel{sec:frequency}{{5}{2}} \newlabel{sec:frequency}{{5}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Further necessary properties of a minimal counterexample}}{3}{}\protected@file@percent }
\newlabel{tocindent-1}{0pt} \newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt} \newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Further necessary properties of a minimal counterexample}}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Edge-deletion subgraphs}}{4}{}\protected@file@percent }
\newlabel{def:edge-deletion}{{7.1}{4}}
\newlabel{thm:edge-deletion-4colorable}{{7.2}{4}}
\gdef \@abspage@last{4} \gdef \@abspage@last{4}
papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk
+5 -3
View File
@@ -1,5 +1,5 @@
# Fdb version 3 # Fdb version 3
["pdflatex"] 1778732276 "paper.tex" "paper.pdf" "paper" 1778732276 ["pdflatex"] 1778732745 "paper.tex" "paper.pdf" "paper" 1778732746
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/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/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/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
@@ -29,9 +29,11 @@
"/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/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/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/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
"/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/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/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/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c ""
"/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/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/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/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 ""
@@ -120,8 +122,8 @@
"/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/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-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
"paper.aux" 1778732276 1234 809d17d72706986551d4267509a33a08 "pdflatex" "paper.aux" 1778732746 1438 b028da99404765220b31b8b219d41cca "pdflatex"
"paper.tex" 1778732267 11154 4367545e63b37b360cb8c3c1c287d046 "" "paper.tex" 1778732735 12434 2966cec5ecd6b41acdbce2ab2207a710 ""
(generated) (generated)
"paper.aux" "paper.aux"
"paper.log" "paper.log"
papers/flip_symmetric_maximal_planar_graphs/paper.fls
+2
View File
@@ -445,6 +445,8 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf
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/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/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/cmsy10.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/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/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/cmtt10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb
papers/flip_symmetric_maximal_planar_graphs/paper.log
+13 -11
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) 14 MAY 2026 00:17 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 00:25
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -504,10 +504,10 @@ hs \OT1/cmr/m/n/10 with \OT1/cmtt/m/n/10 minimum[]connectivity $\OT1/cmr/m/n/10
[2] [3] [4] (./paper.aux) ) [2] [3] [4] (./paper.aux) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
13153 strings out of 478268 13155 strings out of 478268
265523 string characters out of 5846347 265572 string characters out of 5846347
542761 words of memory out of 5000000 542781 words of memory out of 5000000
30991 multiletter control sequences out of 15000+600000 30993 multiletter control sequences out of 15000+600000
475834 words of font info for 54 fonts, out of 8000000 for 9000 475834 words of font info for 54 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191 1302 hyphenation exceptions out of 8191
100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s 100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s
@@ -521,13 +521,15 @@ sfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
onts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont onts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c s/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
m/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm m/cmsy10.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/ /cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm mti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
tt10.pfb> ti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt
Output written on paper.pdf (4 pages, 165847 bytes). 10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/
msam10.pfb>
Output written on paper.pdf (4 pages, 186074 bytes).
PDF statistics: PDF statistics:
80 PDF objects out of 1000 (max. 8388607) 90 PDF objects out of 1000 (max. 8388607)
49 compressed objects within 1 object stream 55 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000) 0 named destinations out of 1000 (max. 500000)
13 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)
papers/flip_symmetric_maximal_planar_graphs/paper.pdf
BIN
View File
Binary file not shown.
papers/flip_symmetric_maximal_planar_graphs/paper.tex
+29
View File
@@ -289,6 +289,35 @@ to identifying further necessary properties of a minimum-order
$5$-chromatic maximal planar graph, of which flip-asymmetry is the $5$-chromatic maximal planar graph, of which flip-asymmetry is the
first. first.
\section{Edge-deletion subgraphs}
\begin{definition}[Edge-deletion subgraph]\label{def:edge-deletion}
Let $G$ be a maximal planar graph and $uv \in E(G)$. The
\emph{edge-deletion subgraph at $uv$} is the spanning subgraph
$G - uv = (V(G),\,E(G) \setminus \{uv\})$. Write
$\mathcal{D}(G) = \{G - uv : uv \in E(G)\}$.
\end{definition}
\begin{theorem}\label{thm:edge-deletion-4colorable}
Let $G_0$ be a maximal planar graph of minimum order with
$\chi(G_0) \geq 5$. Then every $H \in \mathcal{D}(G_0)$ is
$4$-colorable.
\end{theorem}
\begin{proof}
Fix $uv \in E(G_0)$ and let $G_0 / uv$ denote the simple planar graph
obtained by contracting $uv$ and discarding parallel edges. Since
$|V(G_0/uv)| = |V(G_0)| - 1$, the minimality of $G_0$ supplies a
proper $4$-coloring $c$ of $G_0 / uv$. Let $z$ be the contracted
vertex and define $c'\colon V(G_0) \to \{1,2,3,4\}$ by
$c'(u) = c'(v) = c(z)$ and $c'(y) = c(y)$ for $y \notin \{u, v\}$.
Every edge of $G_0 - uv$ is either disjoint from $\{u, v\}$ or
incident to exactly one of them; in either case the corresponding
edge of $G_0 / uv$ has distinct endpoints under $c$, so $c'$ assigns
its endpoints distinct colors. The edge $uv$ itself is absent from
$G_0 - uv$, so $c'$ is a proper $4$-coloring of $G_0 - uv$.
\end{proof}
\end{document} \end{document}
%----------------------------------------------------------------------- %-----------------------------------------------------------------------