dual_decomposition: abstract + intro, rename to "edge suppression", bib
- Fill in the abstract and add a Section 1 introduction. - Rename "cubic-graph edge contraction" -> "edge suppression" throughout (section, definition, theorem, captions, prose, labels). The PNG filenames keep their old paths and still resolve. - Reframe edge suppression as a classical operation we recall, not a new concept we introduce; the face-monochromatic-pair conjecture (with its strengthening) is the sole contribution. - Add a bibliography citing Appel-Haken Parts I and II, Robertson-Sanders- Seymour-Thomas, and Gonthier, and \cite them in the intro. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -1,35 +1,51 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The minimal counterexample}}{1}{}\protected@file@percent }
|
||||
\newlabel{lem:triangulate}{{1.1}{1}}
|
||||
\newlabel{def:minimal}{{1.2}{1}}
|
||||
\newlabel{lem:mindeg}{{1.4}{1}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The reduced dual}}{2}{}\protected@file@percent }
|
||||
\newlabel{def:reduced-dual}{{2.1}{2}}
|
||||
\newlabel{def:edge-names}{{2.3}{2}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 2.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:reduced-dual-steps}{{1}{3}}
|
||||
\newlabel{lem:pentagonal-externals}{{2.4}{3}}
|
||||
\newlabel{lem:chord-apex}{{2.6}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 2.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 2.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{5}{}\protected@file@percent }
|
||||
\newlabel{fig:chord-apex-proof}{{2}{5}}
|
||||
\newlabel{lem:kempe-spike}{{2.7}{6}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Cubic-graph edge contraction}}{7}{}\protected@file@percent }
|
||||
\newlabel{def:cubic-edge-contraction}{{3.1}{7}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.1\hbox {}). Left: a fragment of a cubic plane graph with the contracted edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{7}{}\protected@file@percent }
|
||||
\newlabel{fig:cubic-edge-contraction}{{3}{7}}
|
||||
\newlabel{thm:cubic-contraction-4face}{{3.2}{7}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{8}{}\protected@file@percent }
|
||||
\newlabel{sec:toward-4ct}{{4}{8}}
|
||||
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{4.1}{8}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the contracted graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the contraction).}}{9}{}\protected@file@percent }
|
||||
\newlabel{fig:thm-cubic-contraction-4face}{{4}{9}}
|
||||
\newlabel{rem:conj-3-6-empirical}{{4.2}{9}}
|
||||
\newlabel{conj:face-monochromatic-pair-strengthened}{{4.3}{10}}
|
||||
\newlabel{rem:conj-3-8-empirical}{{4.4}{10}}
|
||||
\citation{AH77a}
|
||||
\citation{AHK77}
|
||||
\citation{RSST97}
|
||||
\citation{Gonthier08}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
|
||||
\newlabel{sec:intro}{{1}{1}}
|
||||
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Organization.}}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Companion paper.}}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The minimal counterexample}}{2}{}\protected@file@percent }
|
||||
\newlabel{sec:minimal}{{2}{2}}
|
||||
\newlabel{lem:triangulate}{{2.1}{2}}
|
||||
\newlabel{def:minimal}{{2.2}{2}}
|
||||
\newlabel{lem:mindeg}{{2.4}{3}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The reduced dual}}{3}{}\protected@file@percent }
|
||||
\newlabel{sec:reduced-dual}{{3}{3}}
|
||||
\newlabel{def:reduced-dual}{{3.1}{3}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:reduced-dual-steps}{{1}{4}}
|
||||
\newlabel{def:edge-names}{{3.3}{4}}
|
||||
\newlabel{lem:pentagonal-externals}{{3.4}{4}}
|
||||
\newlabel{lem:chord-apex}{{3.6}{5}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 3.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{6}{}\protected@file@percent }
|
||||
\newlabel{fig:chord-apex-proof}{{2}{6}}
|
||||
\newlabel{lem:kempe-spike}{{3.7}{7}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Edge suppression}}{8}{}\protected@file@percent }
|
||||
\newlabel{sec:edge-suppression}{{4}{8}}
|
||||
\newlabel{def:edge-suppression}{{4.1}{8}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{8}{}\protected@file@percent }
|
||||
\newlabel{fig:edge-suppression}{{3}{8}}
|
||||
\newlabel{thm:edge-suppression-4face}{{4.2}{8}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{9}{}\protected@file@percent }
|
||||
\newlabel{sec:toward-4ct}{{5}{9}}
|
||||
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{9}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the suppressed graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the suppression).}}{10}{}\protected@file@percent }
|
||||
\newlabel{fig:thm-edge-suppression-4face}{{4}{10}}
|
||||
\newlabel{rem:conj-3-6-empirical}{{5.2}{10}}
|
||||
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.3}{11}}
|
||||
\newlabel{rem:conj-3-8-empirical}{{5.4}{11}}
|
||||
\bibcite{AH77a}{1}
|
||||
\bibcite{AHK77}{2}
|
||||
\bibcite{RSST97}{3}
|
||||
\bibcite{Gonthier08}{4}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\newlabel{rem:implication-4ct}{{4.5}{11}}
|
||||
\gdef \@abspage@last{11}
|
||||
\newlabel{rem:implication-4ct}{{5.5}{12}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{12}
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:15
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:32
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -192,121 +192,128 @@ 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}]
|
||||
Overfull \hbox (41.917pt too wide) in paragraph at lines 145--147
|
||||
[2]
|
||||
Overfull \hbox (41.917pt too wide) in paragraph at lines 233--235
|
||||
[]\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun
|
||||
d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
|
||||
(\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$.
|
||||
[]
|
||||
|
||||
<fig_reduced_dual_step1.png, id=17, 517.79329pt x 499.08812pt>
|
||||
<fig_reduced_dual_step1.png, id=28, 517.79329pt x 499.08812pt>
|
||||
File: fig_reduced_dual_step1.png Graphic file (type png)
|
||||
<use fig_reduced_dual_step1.png>
|
||||
Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 165.
|
||||
Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 253.
|
||||
(pdftex.def) Requested size: 172.79846pt x 166.55775pt.
|
||||
<fig_reduced_dual_step2.png, id=19, 490.16064pt x 483.35876pt>
|
||||
<fig_reduced_dual_step2.png, id=30, 490.16064pt x 483.35876pt>
|
||||
File: fig_reduced_dual_step2.png Graphic file (type png)
|
||||
<use fig_reduced_dual_step2.png>
|
||||
Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 166.
|
||||
Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 254.
|
||||
(pdftex.def) Requested size: 172.79846pt x 170.39505pt.
|
||||
<fig_reduced_dual_step3.png, id=20, 490.16064pt x 483.35876pt>
|
||||
<fig_reduced_dual_step3.png, id=31, 490.16064pt x 483.35876pt>
|
||||
File: fig_reduced_dual_step3.png Graphic file (type png)
|
||||
<use fig_reduced_dual_step3.png>
|
||||
Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 167.
|
||||
Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 255.
|
||||
(pdftex.def) Requested size: 172.79846pt x 170.39505pt.
|
||||
<fig_reduced_dual_step4.png, id=21, 490.16064pt x 486.3346pt>
|
||||
<fig_reduced_dual_step4.png, id=32, 490.16064pt x 486.3346pt>
|
||||
File: fig_reduced_dual_step4.png Graphic file (type png)
|
||||
<use fig_reduced_dual_step4.png>
|
||||
Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 168.
|
||||
Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 256.
|
||||
(pdftex.def) Requested size: 172.79846pt x 171.44409pt.
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
[2] [3 <./fig_reduced_dual_step1.png> <./fig_reduced_dual_step2.png> <./fig_red
|
||||
[3] [4 <./fig_reduced_dual_step1.png> <./fig_reduced_dual_step2.png> <./fig_red
|
||||
uced_dual_step3.png> <./fig_reduced_dual_step4.png>]
|
||||
<fig_chord_apex_step1.png, id=34, 505.03976pt x 502.06393pt>
|
||||
<fig_chord_apex_step1.png, id=43, 505.03976pt x 502.06393pt>
|
||||
File: fig_chord_apex_step1.png Graphic file (type png)
|
||||
<use fig_chord_apex_step1.png>
|
||||
Package pdftex.def Info: fig_chord_apex_step1.png used on input line 289.
|
||||
Package pdftex.def Info: fig_chord_apex_step1.png used on input line 377.
|
||||
(pdftex.def) Requested size: 251.9989pt x 250.5104pt.
|
||||
<fig_chord_apex_step2.png, id=35, 490.16064pt x 499.51323pt>
|
||||
<fig_chord_apex_step2.png, id=44, 490.16064pt x 499.51323pt>
|
||||
File: fig_chord_apex_step2.png Graphic file (type png)
|
||||
<use fig_chord_apex_step2.png>
|
||||
Package pdftex.def Info: fig_chord_apex_step2.png used on input line 290.
|
||||
Package pdftex.def Info: fig_chord_apex_step2.png used on input line 378.
|
||||
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
|
||||
<fig_chord_apex_step3.png, id=36, 490.16064pt x 499.51323pt>
|
||||
<fig_chord_apex_step3.png, id=45, 490.16064pt x 499.51323pt>
|
||||
File: fig_chord_apex_step3.png Graphic file (type png)
|
||||
<use fig_chord_apex_step3.png>
|
||||
Package pdftex.def Info: fig_chord_apex_step3.png used on input line 291.
|
||||
Package pdftex.def Info: fig_chord_apex_step3.png used on input line 379.
|
||||
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
|
||||
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
|
||||
pex_step3.png>] [6]
|
||||
<fig_cubic_edge_contraction.png, id=49, 950.752pt x 203.159pt>
|
||||
[5] [6 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
|
||||
pex_step3.png>] [7]
|
||||
<fig_cubic_edge_contraction.png, id=59, 950.752pt x 203.159pt>
|
||||
File: fig_cubic_edge_contraction.png Graphic file (type png)
|
||||
<use fig_cubic_edge_contraction.png>
|
||||
Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 432
|
||||
Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 522
|
||||
.
|
||||
(pdftex.def) Requested size: 341.9989pt x 73.08138pt.
|
||||
[7 <./fig_cubic_edge_contraction.png>]
|
||||
<fig_thm_cubic_contraction_4face.png, id=55, 916.223pt x 417.56pt>
|
||||
[8 <./fig_cubic_edge_contraction.png>]
|
||||
<fig_thm_cubic_contraction_4face.png, id=64, 916.223pt x 417.56pt>
|
||||
File: fig_thm_cubic_contraction_4face.png Graphic file (type png)
|
||||
<use fig_thm_cubic_contraction_4face.png>
|
||||
Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin
|
||||
e 507.
|
||||
e 597.
|
||||
(pdftex.def) Requested size: 352.79846pt x 160.78339pt.
|
||||
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
[8]
|
||||
[9]
|
||||
Underfull \vbox (badness 10000) has occurred while \output is active []
|
||||
|
||||
[9 <./fig_thm_cubic_contraction_4face.png>]
|
||||
Underfull \hbox (badness 1648) in paragraph at lines 638--644
|
||||
\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 4.4\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
|
||||
[10 <./fig_thm_cubic_contraction_4face.png>]
|
||||
Underfull \hbox (badness 1648) in paragraph at lines 721--727
|
||||
\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.4\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
|
||||
he strength-ened con-jec-ture was tested on the same chord-
|
||||
[]
|
||||
|
||||
|
||||
Underfull \hbox (badness 1014) in paragraph at lines 638--644
|
||||
\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 4.2[]; for each colour-ing we
|
||||
Underfull \hbox (badness 1014) in paragraph at lines 721--727
|
||||
\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.2[]; for each colour-ing we
|
||||
sought any
|
||||
[]
|
||||
|
||||
[10] [11] (./paper.aux) )
|
||||
[11] [12] (./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
3087 strings out of 478268
|
||||
44184 string characters out of 5846347
|
||||
350310 words of memory out of 5000000
|
||||
21121 multiletter control sequences out of 15000+600000
|
||||
478077 words of font info for 62 fonts, out of 8000000 for 9000
|
||||
3097 strings out of 478268
|
||||
44287 string characters out of 5846347
|
||||
350362 words of memory out of 5000000
|
||||
21130 multiletter control sequences out of 15000+600000
|
||||
478386 words of font info for 63 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,9n,76p,744b,373s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
69i,9n,76p,745b,389s 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/a
|
||||
msfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
|
||||
sfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
|
||||
onts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
|
||||
ts/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/cm
|
||||
r9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy1
|
||||
0.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/cmsy9.pf
|
||||
b></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/cmtt10.pfb></
|
||||
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pf
|
||||
b>
|
||||
Output written on paper.pdf (11 pages, 941265 bytes).
|
||||
amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
|
||||
sfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
|
||||
sfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
|
||||
fonts/cmextra/cmex8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
|
||||
amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
|
||||
msfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
|
||||
fonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
|
||||
nts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
|
||||
s/cm/cmmi9.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
|
||||
/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
|
||||
r7.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/cmr9.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/cmsy6.pfb></
|
||||
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></us
|
||||
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr/
|
||||
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr/lo
|
||||
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/loc
|
||||
al/texlive/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/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (12 pages, 1015233 bytes).
|
||||
PDF statistics:
|
||||
144 PDF objects out of 1000 (max. 8388607)
|
||||
78 compressed objects within 1 object stream
|
||||
183 PDF objects out of 1000 (max. 8388607)
|
||||
101 compressed objects within 2 object streams
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
46 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -45,12 +45,99 @@ graph, edge connectivity, cyclic edge cut, Tait colouring, $3$-edge-colouring}
|
||||
\dedicatory{}
|
||||
|
||||
\begin{abstract}
|
||||
% TODO: abstract.
|
||||
We propose the \emph{face-monochromatic-pair conjecture}, a structural
|
||||
property of proper $3$-edge-colourings of cubic plane graphs that, if
|
||||
true, implies the Four Colour Theorem. Working in the planar dual $G'$ of
|
||||
a hypothetical minimal counterexample $G$ to 4CT, we delete a single
|
||||
pentagonal face of $G'$ and rewire its five external vertices around a
|
||||
new apex vertex and a chord; the resulting \emph{reduced dual}
|
||||
$\widehat{G}'_{v,i}$ is a smaller cubic plane graph whose proper
|
||||
$3$-edge-colourings, by the minimality of $G$, are constrained by a
|
||||
chord-apex condition and a pair of Kempe-cycle conditions. The
|
||||
face-monochromatic-pair conjecture, in its strengthened form, asserts the
|
||||
existence in every such colouring of a face $F$ and two non-incident
|
||||
same-coloured edges $e_1, e_2 \in \partial F$ whose subdivision-and-%
|
||||
bridging produces a $4$-face $f_n$ whose boundary colouring places it
|
||||
under the hypothesis of a $4$-face edge-suppression theorem; we use this
|
||||
theorem to derive a proper $3$-edge-colouring of $G'$, contradicting
|
||||
minimality. We verify the conjecture computationally on all
|
||||
chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 20$
|
||||
($142{,}812$ colourings, all pass); the weaker form is verified up to
|
||||
$|V(G)| \leq 21$ ($535{,}182$ colourings, all pass).
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Introduction}
|
||||
\label{sec:intro}
|
||||
|
||||
The Four Colour Theorem (4CT) --- that every loopless plane graph admits a
|
||||
proper $4$-vertex-colouring --- has, since the late 1970s, only been proved
|
||||
by computer-assisted case analysis on a discharging argument over a finite
|
||||
set of unavoidable reducible configurations. Appel and Haken's original
|
||||
proof~\cite{AH77a, AHK77}, the Robertson--Sanders--Seymour--Thomas
|
||||
reworking~\cite{RSST97}, and Gonthier's machine-checked
|
||||
version~\cite{Gonthier08} all share that structure.
|
||||
|
||||
This paper takes a different approach: rather than discharge over
|
||||
configurations in the triangulation $G$, we work in its planar dual $G'$,
|
||||
a cubic plane graph whose proper $3$-edge-colourings correspond by Tait's
|
||||
theorem to proper $4$-vertex-colourings of $G$. Assuming $G$ is a minimal
|
||||
counterexample to 4CT, we delete a single pentagonal face of $G'$ and
|
||||
rewire its five external vertices, obtaining a smaller cubic plane graph
|
||||
$\widehat{G}'_{v,i}$ --- the \emph{reduced dual} --- which by minimality
|
||||
\emph{is} properly $3$-edge-colourable. Two structural lemmas constrain
|
||||
every such colouring: a \emph{chord-apex} condition (Lemma~\ref{lem:chord-apex})
|
||||
forcing two named edges to share a colour, and a pair of
|
||||
\emph{Kempe-cycle} conditions (Lemma~\ref{lem:kempe-spike}) placing four
|
||||
of the rewired edges on common bichromatic Kempe cycles. These constraints
|
||||
are the starting point of the present development.
|
||||
|
||||
The main contribution of the paper is the
|
||||
\emph{face-monochromatic-pair conjecture}
|
||||
(Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}) and
|
||||
its strengthening
|
||||
(Conjecture~\ref{conj:face-monochromatic-pair-strengthened}), which we
|
||||
show together imply the Four Colour Theorem. The supporting ingredients
|
||||
are the chord-apex and Kempe-cycle lemmas on reduced-dual colourings, the
|
||||
classical operation of \emph{edge suppression} (delete the edge and smooth
|
||||
its two degree-$2$ endpoints; equivalently, simple-graph contraction in
|
||||
the dual triangulation; recalled in Section~\ref{sec:edge-suppression}),
|
||||
and an observation that suppression preserves $3$-edge-colourability when
|
||||
applied across a $4$-face whose two opposite boundary edges carry
|
||||
different colours (Theorem~\ref{thm:edge-suppression-4face}).
|
||||
|
||||
The strategy is to construct, from a putative minimal counterexample's
|
||||
reduced-dual colouring, a $4$-face $f_n$ in a slightly modified graph
|
||||
$\widehat{G}'^{+}$ to which the suppression theorem applies; the
|
||||
suppression then produces a properly $3$-edge-coloured graph from which a
|
||||
$3$-edge-colouring of $G'$ can be recovered, contradicting the
|
||||
non-$4$-colourability of $G$. The face-monochromatic-pair conjecture
|
||||
asserts the existence of the structural data ($F, e_1, e_2$) needed to
|
||||
build $f_n$; the strengthening guarantees that $f_n$'s boundary colouring
|
||||
falls under the suppression theorem's hypothesis. Both conjectures have
|
||||
been verified computationally on all chord-apex+Kempe colourings of
|
||||
reduced duals up to $|V(G)| \leq 20$, with the weaker form going up to
|
||||
$|V(G)| \leq 21$.
|
||||
|
||||
\paragraph{Organization.} Section~\ref{sec:minimal} fixes the
|
||||
minimal-counterexample framework: $G$ is a triangulation,
|
||||
$\delta(G) \geq 5$, and every triangulation on fewer vertices is properly
|
||||
$4$-colourable. Section~\ref{sec:reduced-dual} introduces the reduced dual
|
||||
$\widehat{G}'_{v,i}$ and proves the chord-apex and Kempe-cycle lemmas.
|
||||
Section~\ref{sec:edge-suppression} defines edge suppression and proves
|
||||
its $4$-face $3$-edge-colourability theorem.
|
||||
Section~\ref{sec:toward-4ct} states the two conjectures, reports the
|
||||
empirical verification, and gives the implication to 4CT.
|
||||
|
||||
\paragraph{Companion paper.} An iterated version of the reduced-dual
|
||||
construction --- producing a sequence $H_1, H_2, \dots$ of progressively
|
||||
smaller cubic plane graphs and tracking an accumulating ``protected'' edge
|
||||
set --- is the subject of a companion paper. The present paper does not
|
||||
use that iteration.
|
||||
|
||||
\section{The minimal counterexample}
|
||||
\label{sec:minimal}
|
||||
|
||||
Throughout, a \emph{triangulation} is a simple plane graph, with a fixed
|
||||
embedding, in which every face --- including the outer face --- is bounded by a
|
||||
@@ -118,6 +205,7 @@ Hence $\delta(G) \ge 5$.
|
||||
\end{proof}
|
||||
|
||||
\section{The reduced dual}
|
||||
\label{sec:reduced-dual}
|
||||
|
||||
Write $G'$ for the dual of $G$: since $G$ is a triangulation, $G'$ is a cubic
|
||||
plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
|
||||
@@ -393,61 +481,63 @@ distinct colours, contradicting Lemma~\ref{lem:chord-apex} applied to
|
||||
$\varphi'$.
|
||||
\end{proof}
|
||||
|
||||
\section{Cubic-graph edge contraction}
|
||||
\section{Edge suppression}
|
||||
\label{sec:edge-suppression}
|
||||
|
||||
We introduce a cubic-preserving analogue of edge contraction which turns
|
||||
out --- under planar duality --- to coincide with simple-graph contraction
|
||||
on the dual side. It will be the central tool in
|
||||
We recall the classical operation of \emph{edge suppression} on cubic
|
||||
plane graphs: delete the edge and smooth the two resulting degree-$2$
|
||||
endpoints. Under planar duality this coincides with simple-graph
|
||||
contraction on the dual side. It will be the central tool in
|
||||
Section~\ref{sec:toward-4ct} below, where we formulate a sufficient
|
||||
condition for the Four Colour Theorem.
|
||||
|
||||
\begin{definition}[Cubic-graph edge contraction]
|
||||
\label{def:cubic-edge-contraction}
|
||||
Let $H$ be a cubic plane graph and $e = uv$ an edge of $H$ with $u \neq v$ and
|
||||
no edge of $H$ parallel to $e$. The \emph{cubic-graph edge contraction} of $H$
|
||||
along $e$ is the graph $H'$ obtained in two steps:
|
||||
\begin{definition}[Edge suppression]
|
||||
\label{def:edge-suppression}
|
||||
Let $H$ be a cubic plane graph and $e = uv$ an edge of $H$ with $u \neq v$
|
||||
and no edge of $H$ parallel to $e$. The \emph{edge suppression} of $H$ at
|
||||
$e$ is the graph $H'$ obtained in two steps:
|
||||
\begin{enumerate}
|
||||
\item \emph{Delete} the edge $e$; the endpoints $u$ and $v$ each drop to
|
||||
degree $2$.
|
||||
\item \emph{Smooth} each of $u$ and $v$: at $u$, replace $u$ and its two
|
||||
remaining incident edges $ua, ub$ by a single new edge $ab$; do the
|
||||
same at $v$. Both vertices $u$ and $v$ are removed, and two new edges
|
||||
are added in their place.
|
||||
remaining incident edges $ua, ub$ by a single new edge $ab$; do
|
||||
the same at $v$. Both vertices $u$ and $v$ are removed, and two
|
||||
new edges are added in their place.
|
||||
\end{enumerate}
|
||||
Provided the smoothings do not introduce a loop or parallel edge, $H'$ is
|
||||
again a cubic plane graph, with $|V(H')| = |V(H)| - 2$ and
|
||||
$|E(H')| = |E(H)| - 3$.
|
||||
|
||||
Equivalently, $H'$ is the planar dual of $\mathrm{dual}(H) / e^{*}$, where
|
||||
$e^{*}$ is the edge of $\mathrm{dual}(H)$ crossing $e$ and the contraction
|
||||
on the right-hand side is simple-graph contraction (loops removed, parallel
|
||||
edges absorbed). Under planar duality, contracting $e^{*}$ in
|
||||
$\mathrm{dual}(H)$ merges the two triangular faces of $\mathrm{dual}(H)$
|
||||
incident to $e^{*}$, and the parallel-edge cleanup corresponds exactly to
|
||||
the smoothing step on the primal side.
|
||||
Equivalently, $H'$ is the planar dual of $\mathrm{dual}(H) / e^{*}$,
|
||||
where $e^{*}$ is the edge of $\mathrm{dual}(H)$ crossing $e$ and the
|
||||
contraction on the right-hand side is simple-graph contraction (loops
|
||||
removed, parallel edges absorbed). Under planar duality, contracting
|
||||
$e^{*}$ in $\mathrm{dual}(H)$ merges the two triangular faces of
|
||||
$\mathrm{dual}(H)$ incident to $e^{*}$, and the parallel-edge cleanup
|
||||
corresponds exactly to the smoothing step on the primal side.
|
||||
\end{definition}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.95\textwidth]{fig_cubic_edge_contraction.png}
|
||||
\caption{Cubic-graph edge contraction
|
||||
(Definition~\ref{def:cubic-edge-contraction}). Left: a fragment of a cubic
|
||||
plane graph with the contracted edge $e = uv$ highlighted in red. Middle:
|
||||
deleting $e$ leaves $u$ and $v$ of degree~$2$. Right: smoothing $u$ and $v$
|
||||
replaces each pair of incident edges by a single new edge, removing $u, v$
|
||||
and giving a cubic plane graph again.}
|
||||
\label{fig:cubic-edge-contraction}
|
||||
\caption{Edge suppression
|
||||
(Definition~\ref{def:edge-suppression}). Left: a fragment of a cubic
|
||||
plane graph with the suppressed edge $e = uv$ highlighted in red. Middle:
|
||||
deleting $e$ leaves $u$ and $v$ of degree~$2$. Right: smoothing $u$ and
|
||||
$v$ replaces each pair of incident edges by a single new edge, removing
|
||||
$u, v$ and giving a cubic plane graph again.}
|
||||
\label{fig:edge-suppression}
|
||||
\end{figure}
|
||||
|
||||
\begin{theorem}[Cubic contraction across a 4-face preserves 3-edge-colourability]
|
||||
\label{thm:cubic-contraction-4face}
|
||||
\begin{theorem}[Edge suppression across a 4-face preserves 3-edge-colourability]
|
||||
\label{thm:edge-suppression-4face}
|
||||
Let $H$ be a cubic plane graph with a proper $3$-edge-colouring $\varphi$,
|
||||
let $f$ be a face of $H$ with $|\partial f| = 4$, and let $e_0, e_1$ be the
|
||||
two edges of $\partial f$ sharing no endpoint (the opposite pair on the
|
||||
$4$-cycle $\partial f$). If $\varphi(e_0) \neq \varphi(e_1)$ and the
|
||||
cubic-graph edge contraction of $H$ along $e_0$
|
||||
(Definition~\ref{def:cubic-edge-contraction}) is well-defined (no loops or
|
||||
parallel edges are created), then the contracted graph admits a proper
|
||||
let $f$ be a face of $H$ with $|\partial f| = 4$, and let $e_0, e_1$ be
|
||||
the two edges of $\partial f$ sharing no endpoint (the opposite pair on
|
||||
the $4$-cycle $\partial f$). If $\varphi(e_0) \neq \varphi(e_1)$ and the
|
||||
edge suppression of $H$ at $e_0$
|
||||
(Definition~\ref{def:edge-suppression}) is well-defined (no loops or
|
||||
parallel edges are created), then the suppressed graph admits a proper
|
||||
$3$-edge-colouring.
|
||||
\end{theorem}
|
||||
|
||||
@@ -466,8 +556,8 @@ $\varphi(w_1) = b$ and $\varphi(w_2) = a$. Symmetrically $\varphi(e_3) = c$,
|
||||
$\varphi(w_0) = b$, and $\varphi(w_3) = a$. In particular
|
||||
$\varphi(w_0) = \varphi(w_1) = b$.
|
||||
|
||||
\emph{Construction of $\varphi'$.} Let $H'$ denote the cubic-graph edge
|
||||
contraction of $H$ along $e_0$; its new edges are $e_3' := v_3 u_0$
|
||||
\emph{Construction of $\varphi'$.} Let $H'$ denote the edge suppression
|
||||
of $H$ at $e_0$; its new edges are $e_3' := v_3 u_0$
|
||||
(replacing $e_3$ and $w_0$ via the smoothing at $v_0$) and
|
||||
$e_2' := v_2 u_1$ (replacing $e_2$ and $w_1$ via the smoothing at $v_1$).
|
||||
Define $\varphi' \colon E(H') \to \{1, 2, 3\}$ by
|
||||
@@ -506,16 +596,16 @@ $\varphi'$ is proper.
|
||||
\centering
|
||||
\includegraphics[width=0.98\textwidth]{fig_thm_cubic_contraction_4face.png}
|
||||
\caption{The recolouring used in the proof of
|
||||
Theorem~\ref{thm:cubic-contraction-4face}. Left: the $4$-face $f$ of $H$
|
||||
Theorem~\ref{thm:edge-suppression-4face}. Left: the $4$-face $f$ of $H$
|
||||
under $\varphi$, with the forced colours $\varphi(e_0) = a$,
|
||||
$\varphi(e_1) = b$, $\varphi(e_2) = \varphi(e_3) = c$,
|
||||
$\varphi(w_0) = \varphi(w_1) = b$, and $\varphi(w_2) = \varphi(w_3) = a$.
|
||||
Right: the contracted graph $H'$ under $\varphi'$. The smoothed-in edges
|
||||
Right: the suppressed graph $H'$ under $\varphi'$. The smoothed-in edges
|
||||
$e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is
|
||||
recoloured from $b$ to $c$; every edge outside the face neighbourhood
|
||||
keeps its $\varphi$-colour (dotted in red: the five edges of $H$ removed
|
||||
by the contraction).}
|
||||
\label{fig:thm-cubic-contraction-4face}
|
||||
by the suppression).}
|
||||
\label{fig:thm-edge-suppression-4face}
|
||||
\end{figure}
|
||||
|
||||
\section{The face-monochromatic-pair conjecture and the Four Colour Theorem}
|
||||
@@ -524,14 +614,7 @@ by the contraction).}
|
||||
The following conjecture identifies a structural property of every proper
|
||||
$3$-edge-colouring of a reduced dual of a minimal counterexample. If true,
|
||||
it implies the Four Colour Theorem via
|
||||
Theorem~\ref{thm:cubic-contraction-4face}.
|
||||
|
||||
The reduced-dual construction of Definition~\ref{def:reduced-dual} can also
|
||||
be applied \emph{iteratively}, producing a sequence $H_1, H_2, \dots$ of
|
||||
progressively smaller cubic plane graphs and tracking an accumulating set
|
||||
of ``protected'' edges; that iterated reduction, with its termination
|
||||
analysis and structural lemmas about spike/merged pairs at each stage, is
|
||||
the subject of a companion paper and is not used below.
|
||||
Theorem~\ref{thm:edge-suppression-4face}.
|
||||
|
||||
\begin{conjecture}[Face-monochromatic-pair conjecture]
|
||||
\label{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
||||
@@ -601,7 +684,7 @@ has no content there.
|
||||
|
||||
The following strengthening adds a fourth clause that arranges the new
|
||||
$4$-face $f_n$ to satisfy the hypotheses of
|
||||
Theorem~\ref{thm:cubic-contraction-4face}.
|
||||
Theorem~\ref{thm:edge-suppression-4face}.
|
||||
|
||||
\begin{conjecture}[Strengthening of
|
||||
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}]
|
||||
@@ -674,11 +757,11 @@ carry distinct $\varphi'$-colours, the cyclic colour pattern on
|
||||
$\partial f_n$ must be $(c, a, c, b)$ up to rotation and relabelling, with
|
||||
the two $c$-edges opposite and the two remaining opposite edges carrying
|
||||
the distinct colours $a$ and $b$. Those two opposite edges therefore
|
||||
satisfy the hypothesis of Theorem~\ref{thm:cubic-contraction-4face}: they
|
||||
satisfy the hypothesis of Theorem~\ref{thm:edge-suppression-4face}: they
|
||||
lie on the $4$-face $f_n$, share no endpoint, and have different
|
||||
$\varphi'$-colours. Theorem~\ref{thm:cubic-contraction-4face} then produces
|
||||
a proper $3$-edge-colouring of the cubic-graph edge contraction
|
||||
$\widehat{G}'^{+} \!\setminus\! e$ along the $a$-coloured one.
|
||||
$\varphi'$-colours. Theorem~\ref{thm:edge-suppression-4face} then produces
|
||||
a proper $3$-edge-colouring of the edge suppression of
|
||||
$\widehat{G}'^{+}$ at the $a$-coloured one.
|
||||
|
||||
Case~(ii) of clause~(4) is conjecturally reducible to case~(i) by a single
|
||||
Kempe swap on the $\{b, c\}$-cycle through $X_1 X_2$: by hypothesis that
|
||||
@@ -688,9 +771,9 @@ $\partial f_n$ unchanged --- placing $\partial f_n$ into the three-colour
|
||||
pattern of case~(i).
|
||||
|
||||
\smallskip\noindent\emph{Consequence.}
|
||||
Theorem~\ref{thm:cubic-contraction-4face} now produces a proper
|
||||
$3$-edge-colouring of the cubic-graph edge contraction of
|
||||
$\widehat{G}'^{+}$ along the chosen edge of $f_n$. Combined with the
|
||||
Theorem~\ref{thm:edge-suppression-4face} now produces a proper
|
||||
$3$-edge-colouring of the edge suppression of $\widehat{G}'^{+}$ at the
|
||||
chosen edge of $f_n$. Combined with the
|
||||
chord-apex and Kempe-cycle structure of $\widehat{G}'_{v,i}$
|
||||
(Lemmas~\ref{lem:chord-apex} and~\ref{lem:kempe-spike}), this yields a
|
||||
proper $3$-edge-colouring of $G'$, and by Tait's correspondence a proper
|
||||
@@ -700,4 +783,26 @@ Conjecture~\ref{conj:face-monochromatic-pair-strengthened} implies the
|
||||
Four Colour Theorem.
|
||||
\end{remark}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
\bibitem{AH77a}
|
||||
K.~Appel and W.~Haken,
|
||||
\emph{Every planar map is four colorable. Part~I: Discharging},
|
||||
Illinois J.~Math. \textbf{21} (1977), 429--490.
|
||||
|
||||
\bibitem{AHK77}
|
||||
K.~Appel, W.~Haken, and J.~Koch,
|
||||
\emph{Every planar map is four colorable. Part~II: Reducibility},
|
||||
Illinois J.~Math. \textbf{21} (1977), 491--567.
|
||||
|
||||
\bibitem{RSST97}
|
||||
N.~Robertson, D.~P.~Sanders, P.~Seymour, and R.~Thomas,
|
||||
\emph{The four-colour theorem},
|
||||
J.~Combin. Theory Ser.~B \textbf{70} (1997), 2--44.
|
||||
|
||||
\bibitem{Gonthier08}
|
||||
G.~Gonthier,
|
||||
\emph{Formal proof---the four-color theorem},
|
||||
Notices Amer. Math. Soc. \textbf{55} (2008), no.~11, 1382--1393.
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user