papers: rename folders and retitle

- Main paper: dual_decomposition_minimal_counterexamples/ ->
  face_monochromatic_pairs/. Title is now
  "Face-Monochromatic Pairs and the Four Colour Theorem".
- Companion paper: dual_decomposition_iterated_reduction/ ->
  iterated_reduction_in_reduced_dual/. Title is now
  "An Iterated Reduction in the Reduced Dual". Its prose and bibliography
  cite the parent under the new title.
- Update one absolute sys.path reference inside
  check_conj_face_kempe_n15.py that pointed at the old folder.

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

papers/face_monochromatic_pairs/experiments/check_conj_face_kempe_n15.py

@@ -4,7 +4,7 @@ from sage.graphs.graph_generators import graphs
 import sys
 
 # Reuse helpers from check_conj_face_kempe.py by importing
-sys.path.insert(0, '/Users/didericis/Code/math-research/papers/dual_decomposition_minimal_counterexamples/experiments')
+sys.path.insert(0, '/Users/didericis/Code/math-research/papers/face_monochromatic_pairs/experiments')
 from check_conj_face_kempe import (
     dual_of, apply_reduction, proper_3_edge_colorings,
     kempe_cycle, edge_idx, matches_chord_apex_kempe,

papers/face_monochromatic_pairs/paper.aux

@@ -18,17 +18,17 @@
 \@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:pentagonal-externals}{{3.4}{5}}
+\newlabel{lem:chord-apex}{{3.6}{6}}
+\@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 $G'$.}}{7}{}\protected@file@percent }
+\newlabel{fig:chord-apex-proof}{{2}{7}}
 \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{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.}}{9}{}\protected@file@percent }
+\newlabel{fig:edge-suppression}{{3}{9}}
 \@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}}

papers/face_monochromatic_pairs/paper.log

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papers/face_monochromatic_pairs/paper.tex

@@ -26,7 +26,7 @@
 
 \begin{document}
 
-\title{Dual Decomposition of Minimal Counterexamples}
+\title{Face-Monochromatic Pairs and the Four Colour Theorem}
 
 %    author one information
 \author{Eric Bauerfeld}

papers/iterated_reduction_in_reduced_dual/paper.log

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papers/iterated_reduction_in_reduced_dual/paper.tex

@@ -26,7 +26,7 @@
 
 \begin{document}
 
-\title{Iterated Reduction of Dual Minimal Counterexamples}
+\title{An Iterated Reduction in the Reduced Dual}
 
 \author{Eric Bauerfeld}
 \address{}
@@ -53,8 +53,8 @@ $3$-edge-colouring}
 \section{Setup and background}
 \label{sec:background}
 
-This paper is a follow-up to \emph{Dual Decomposition of Minimal
-Counterexamples}~\cite{parent}, which introduced the reduced-dual
+This paper is a follow-up to \emph{Face-Monochromatic Pairs and the Four
+Colour Theorem}~\cite{parent}, which introduced the reduced-dual
 construction: given a minimal counterexample $G$ to the Four Colour
 Theorem, a degree-$5$ vertex $v$ of $G$ (equivalently a pentagonal face
 $F_v$ of $G' = \mathrm{dual}(G)$), and an index $i \in \{0,1,2,3,4\}$, the
@@ -301,7 +301,7 @@ for every $t \in \{1, \dots, t^*\}$.
 
 \begin{thebibliography}{9}
 \bibitem{parent}
-E.~Bauerfeld, \emph{Dual Decomposition of Minimal Counterexamples}.
+E.~Bauerfeld, \emph{Face-Monochromatic Pairs and the Four Colour Theorem}.
 Companion paper.
 \end{thebibliography}