face_monochromatic_pairs: fix Remark 5.6 formatting

The disproof of Conjecture 5.5 used \[ ... \qquad\text{and}\qquad ... \]
to give both cycle definitions on a single display line. The second
half was too long to fit, producing a wide gap and an ugly line break.
Switch to align* with the two definitions stacked and aligned at =.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
2026-05-25 02:58:16 -04:00
parent cc81b804db
commit 6eb85d220c
2 changed files with 4 additions and 5 deletions
Binary file not shown.
+4 -5
View File
@@ -833,13 +833,12 @@ 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
\[
\begin{align*}
K_{\mathrm{red},\mathrm{blue}}
\;=\; \text{the outer $8$-cycle}
\qquad\text{and}\qquad
&= \text{the outer $8$-cycle, and} \\
K_{\mathrm{red},\mathrm{green}}
\;=\; \text{the $12$-cycle (outer + upper-left ``ladder'' side)}
\]
&= \text{the $12$-cycle (outer frame $+$ upper-left ``ladder'' side)}
\end{align*}
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