Add Kempe-chain structure of 4-colorings of edge-deletion subgraphs

For G_0 a minimum-order 5-chromatic maximal planar graph and any
4-coloring of G_0 - uv, the endpoints u, v must share a color, and the
color classes pairing that color with each of two other colors must
each induce a u-v path. The Kempe-chain parts follow from a standard
swap-on-component contradiction against the shared-color claim.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/flip_symmetric_maximal_planar_graphs/paper.aux

@@ -17,4 +17,5 @@
 \@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}}
+\newlabel{thm:edge-deletion-coloring-structure}{{7.3}{4}}
 \gdef \@abspage@last{4}

papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk

@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1778732745 "paper.tex" "paper.pdf" "paper" 1778732746
+["pdflatex"] 1778733012 "paper.tex" "paper.pdf" "paper" 1778733012
   "/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 ""
@@ -122,8 +122,8 @@
   "/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" 1778732746 1438 b028da99404765220b31b8b219d41cca "pdflatex"
+  "paper.aux" 1778733012 1496 56d68f0294258d5724b6317ee89e6aaa "pdflatex"
-  "paper.tex" 1778732735 12434 2966cec5ecd6b41acdbce2ab2207a710 ""
+  "paper.tex" 1778733006 13839 954cc3cf73e27b836fc30fd8b0d7a40b ""
   (generated)
   "paper.aux"
   "paper.log"

papers/flip_symmetric_maximal_planar_graphs/paper.log

@@ -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 00:25
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  14 MAY 2026 00:30
 entering extended mode
  restricted \write18 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) ) 
 Here is how much of TeX's memory you used:
- 13155 strings out of 478268
+ 13156 strings out of 478268
- 265572 string characters out of 5846347
+ 265610 string characters out of 5846347
- 542781 words of memory out of 5000000
+ 542791 words of memory out of 5000000
- 30993 multiletter control sequences out of 15000+600000
+ 30994 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
  100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s
@@ -526,7 +526,7 @@ mti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
 ti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt
 10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/
 msam10.pfb>
-Output written on paper.pdf (4 pages, 186074 bytes).
+Output written on paper.pdf (4 pages, 189743 bytes).
 PDF statistics:
  90 PDF objects out of 1000 (max. 8388607)
  55 compressed objects within 1 object stream

papers/flip_symmetric_maximal_planar_graphs/paper.tex

@@ -318,6 +318,38 @@ 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}
 
+\begin{theorem}\label{thm:edge-deletion-coloring-structure}
+Let $G_0$ be a maximal planar graph of minimum order with
+$\chi(G_0) \geq 5$, fix $uv \in E(G_0)$, and let $\varphi$ be any
+proper $4$-coloring of $G_0 - uv$.  Write $a = \varphi(u)$ and let
+$b, c, d$ denote the three remaining colors.  Then:
+\begin{enumerate}
+\item $\varphi(v) = a$;
+\item the subgraph of $G_0 - uv$ induced by the vertices of color
+$a$ or $b$ contains a path from $u$ to $v$;
+\item the subgraph of $G_0 - uv$ induced by the vertices of color
+$a$ or $c$ contains a path from $u$ to $v$.
+\end{enumerate}
+\end{theorem}
+
+\begin{proof}
+(1) If $\varphi(v) \neq a$ then $\varphi$ is already a proper
+$4$-coloring of $G_0$, since the only edge of $G_0$ absent from
+$G_0 - uv$ is $uv$ and its endpoints have distinct colors.  This
+contradicts $\chi(G_0) \geq 5$, so $\varphi(v) = a$.
+
+(2) Suppose, for contradiction, that $u$ and $v$ lie in distinct
+connected components of the subgraph of $G_0 - uv$ induced by the
+color classes $a$ and $b$.  Let $C$ be the component containing $u$,
+and define $\varphi'\colon V(G_0) \to \{a,b,c,d\}$ by swapping colors
+$a \leftrightarrow b$ on $C$ and leaving every other vertex
+unchanged.  Then $\varphi'$ is a proper $4$-coloring of $G_0 - uv$
+with $\varphi'(u) = b$ and $\varphi'(v) = a$, contradicting part~(1)
+applied to $\varphi'$.
+
+(3) Identical to (2) with $c$ in place of $b$.
+\end{proof}
+
 \end{document}
 
 %-----------------------------------------------------------------------