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face_monochromatic_pairs: embed counterexample figure into paper

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Replaces the placeholder fbox figure with the rendered PNG from
experiments/counterexample_conj_5_5.py (40-vertex cubic plane graph,
proper 3-edge-coloured, both K_{red,blue} 8-cycle and K_{red,green}
12-cycle constant h_φ = -1, sharing the colour-red edge (0, 7)).

Disproof remark also updated to give the actual structural details
(40 vertices, the +1/-1 split 16/24, location of the +1-region in
the inner ladder) instead of a placeholder description.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-25 02:56:27 -04:00
parent 34141322ce
commit cc81b804db
2 changed files with 36 additions and 24 deletions
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@@ -828,37 +828,49 @@ $V(K_0)$, then $h_\varphi$ is \emph{not} constant on $V(K_1)$.
\begin{remark}[Disproof of
Conjecture~\ref{conj:no-two-constant-kempe-cycles}]
\label{rem:no-two-constant-kempe-cycles-counterexample}
Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}:
there exists a cubic plane graph $H$ with a proper $3$-edge-colouring
$\varphi$ admitting an $\{a, b\}$-Kempe cycle $K_0$ and an
$\{a, c\}$-Kempe cycle $K_1$ which share a colour-$a$ edge and on which
$h_\varphi$ is simultaneously constant on $V(K_0)$ and on $V(K_1)$. A
concrete counterexample is recorded in
Figure~\ref{fig:no-two-constant-kempe-counterexample}.
Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}.
Figure~\ref{fig:no-two-constant-kempe-counterexample} exhibits a
concrete counterexample: a cubic plane graph $H$ on $40$ vertices with
a proper $3$-edge-colouring $\varphi$ (colours red/blue/green) in
which both
\[
K_{\mathrm{red},\mathrm{blue}}
\;=\; \text{the outer $8$-cycle}
\qquad\text{and}\qquad
K_{\mathrm{red},\mathrm{green}}
\;=\; \text{the $12$-cycle (outer + upper-left ``ladder'' side)}
\]
share the colour-red edge $(0, 7)$ and satisfy
$h_\varphi \equiv -1$ on the vertex set of each. Globally $h_\varphi$
takes value $+1$ on $16$ vertices and $-1$ on $24$ vertices, with all
of the $+1$-vertices concentrated in the inner ``tilted ladder''
region, so that both Kempe cycles miss them entirely.
The construction and verification (cubic, planar, proper
$3$-edge-colouring, Kempe-cycle tracing, Heawood-number computation
via the CW rotation at each vertex) are in
\texttt{experiments/counterexample\_conj\_5\_5.py}, with the source
drawing in \texttt{constant\_heawood\_counterexample.tikz}.
The (partial) proof attempt below establishes the constraint
The partial proof attempt below establishes
$|E(K_0) \cap E(K_1)| \geq 2$ (Step~2) and closes the sub-case where
two shared a-edges are consecutive on \emph{both} cycles
(``Case~A''/Step~4), but the general claim fails because in the
counterexample no pair of shared a-edges is consecutive on both
cycles --- the $K_1$-arc between two $K_0$-consecutive shared a-edges
itself passes through other shared a-edges, breaking the lune-face
assumption.
two shared a-edges are consecutive on \emph{both} cycles (``Case~A''
/ Step~4). The general claim fails because in the counterexample no
pair of shared a-edges is consecutive on both cycles --- the
$K_1$-arc between two $K_0$-consecutive shared a-edges itself passes
through other shared a-edges, so the lune-face assumption of Step~4
does not hold.
\end{remark}
\begin{figure}[h]
\centering
% TODO: replace placeholder with the actual counterexample drawing
% (e.g., the whiteboard photo or a TikZ rendering).
\fbox{\parbox{0.7\textwidth}{\centering
\emph{Placeholder for the counterexample figure.}\\[2pt]
A cubic plane graph $H$ with three edge colours
(red, blue, teal) showing two Kempe cycles sharing a colour edge,
on which $h_\varphi$ is simultaneously constant on both cycles.\\
See \texttt{figures/no-two-constant-kempe-counterexample.\{png,pdf\}}
once the image is committed to the repo.}}
\includegraphics[width=0.85\textwidth]{figures/no-two-constant-kempe-counterexample.png}
\caption{Counterexample to
Conjecture~\ref{conj:no-two-constant-kempe-cycles}.}
Conjecture~\ref{conj:no-two-constant-kempe-cycles}: a cubic plane
graph on $40$ vertices with a proper $3$-edge-colouring on which
$h_\varphi$ is simultaneously constant ($\equiv -1$) on the outer
red/blue $8$-cycle and on the red/green $12$-cycle (outer frame plus
the upper-left ladder side), which share the colour-red edge
$(0, 7)$.}
\label{fig:no-two-constant-kempe-counterexample}
\end{figure}