Frame flip-asymmetry as first of further necessary properties
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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@@ -276,6 +276,19 @@ the minimum-degree-$5$ class --- which already contains every
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candidate minimum-order $5$-chromatic graph --- flip-symmetric
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examples become a vanishing fraction.
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\section{Further necessary properties of a minimal counterexample}
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The frequency data of Section~\ref{sec:frequency} look unflattering
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only when flip-symmetry is weighed against the full class of maximal
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planar graphs. The class that actually matters --- minimum-order
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$5$-chromatic triangulations that also resist every Kempe-style
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reduction --- is far thinner, and flip-symmetry may exclude a
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substantially larger fraction of it if the configurations it removes
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overlap those responsible for Kempe reducibility. We therefore turn
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to identifying further necessary properties of a minimum-order
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$5$-chromatic maximal planar graph, of which flip-asymmetry is the
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first.
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\end{document}
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%-----------------------------------------------------------------------
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