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Frame flip-asymmetry as first of further necessary properties

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Adds a transitional section reframing the frequency results: the
relevant class is not all maximal planar graphs but those that resist
Kempe-style reductions, where flip-asymmetry's exclusion may have
real bite. Sets up subsequent development of additional necessary
properties of a minimum-order 5-chromatic counterexample.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-14 00:18:34 -04:00
parent 076137baaa
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5 changed files with 36 additions and 22 deletions
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papers/flip_symmetric_maximal_planar_graphs/paper.aux
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papers/flip_symmetric_maximal_planar_graphs/paper.tex
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@@ -276,6 +276,19 @@ the minimum-degree-$5$ class --- which already contains every
candidate minimum-order $5$-chromatic graph --- flip-symmetric candidate minimum-order $5$-chromatic graph --- flip-symmetric
examples become a vanishing fraction. examples become a vanishing fraction.
\section{Further necessary properties of a minimal counterexample}
The frequency data of Section~\ref{sec:frequency} look unflattering
only when flip-symmetry is weighed against the full class of maximal
planar graphs. The class that actually matters --- minimum-order
$5$-chromatic triangulations that also resist every Kempe-style
reduction --- is far thinner, and flip-symmetry may exclude a
substantially larger fraction of it if the configurations it removes
overlap those responsible for Kempe reducibility. We therefore turn
to identifying further necessary properties of a minimum-order
$5$-chromatic maximal planar graph, of which flip-asymmetry is the
first.
\end{document} \end{document}
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