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Add min-degree-5 flip-symmetry census through n=26

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The unrestricted census suggested flip-symmetry already excludes a
vanishing fraction of maximal planar graphs; this commit re-runs the
same enumeration over the minimum-degree-5 subclass (where any
minimum-order 5-chromatic counterexample must live) to check whether
the restriction tightens the bound. It does not: the density decays
to zero there as well, only at a gentler geometric rate (~0.63 per
step instead of ~0.5).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-14 00:12:45 -04:00
parent 1749f702cf
commit 076137baaa
6 changed files with 63 additions and 16 deletions
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papers/maximal_planar_graph_edge_flipping/paper.aux → papers/flip_symmetric_maximal_planar_graphs/paper.aux
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papers/maximal_planar_graph_edge_flipping/paper.fls → papers/flip_symmetric_maximal_planar_graphs/paper.fls
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PWD /Users/didericis/Code/math-research/papers/maximal_planar_graph_edge_flipping
PWD /Users/didericis/Code/math-research/papers/flip_symmetric_maximal_planar_graphs
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papers/maximal_planar_graph_edge_flipping/paper.pdf → papers/flip_symmetric_maximal_planar_graphs/paper.pdf
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\begin{document}
\title{Maximal Planar Graph Edge Flipping}
\title{Flip Symmetric Maximal Planar Graphs}
% Remove any unused author tags.
@@ -220,14 +220,61 @@ $14$ & $339{,}722$& $6{,}164$ & $0.018144$ \\
From $n = 10$ onward the ratio $F_n / T_n$ decreases by a factor
approaching $1/2$ at each step, suggesting that the density of
flip-symmetric graphs among maximal planar graphs of order $n$ decays
to zero --- empirically at a roughly geometric rate. In particular,
the conclusion of
Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric} is consistent
with the prevailing trend: as $n$ grows, almost every maximal planar
graph on $n$ vertices is already excluded from flip-symmetry on
purely structural grounds, and any putative counterexample to the
Four Color Theorem is forced into a vanishingly small slice of the
class.
to zero --- empirically at a roughly geometric rate. This tempers
the utility of
Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}: although it
guarantees that a minimum-order counterexample to the Four Color
Theorem lies in the complement of $\mathcal{F}$, that complement
already comprises nearly the entire class of maximal planar graphs
on $n$ vertices once $n$ is moderately large. The structural
exclusion offered by flip-symmetry therefore prunes a vanishingly
small portion of the search space, and this property is unlikely on
its own to be a productive avenue for narrowing the search for a
counterexample.
A natural follow-up question is whether the picture improves when one
restricts attention to maximal planar graphs of minimum degree at
least~$5$, the class to which any minimum-order $5$-chromatic graph
necessarily belongs (a vertex of degree at most~$4$ admits a standard
Kempe reduction). Writing $T^{(5)}_n$ and $F^{(5)}_n$ for the
analogous counts within this subclass, we ran the same census after
adding \texttt{minimum\_degree}~$=5$ to the \texttt{plantri}
invocation, obtaining the table below.
\begin{center}
\begin{tabular}{r r r l}
\hline
$n$ & $T^{(5)}_n$ & $F^{(5)}_n$ & $F^{(5)}_n / T^{(5)}_n$ \\
\hline
$12$ & $1$ & $0$ & $0.000000$ \\
$13$ & $0$ & $0$ & --- \\
$14$ & $1$ & $0$ & $0.000000$ \\
$15$ & $1$ & $0$ & $0.000000$ \\
$16$ & $3$ & $1$ & $0.333333$ \\
$17$ & $4$ & $1$ & $0.250000$ \\
$18$ & $12$ & $2$ & $0.166667$ \\
$19$ & $23$ & $5$ & $0.217391$ \\
$20$ & $73$ & $12$ & $0.164384$ \\
$21$ & $192$ & $27$ & $0.140625$ \\
$22$ & $651$ & $51$ & $0.078341$ \\
$23$ & $2{,}070$ & $120$ & $0.057971$ \\
$24$ & $7{,}290$ & $273$ & $0.037449$ \\
$25$ & $25{,}381$ & $598$ & $0.023561$ \\
$26$ & $91{,}441$ & $1{,}341$ & $0.014665$ \\
\hline
\end{tabular}
\end{center}
The first flip-symmetric example in this subclass appears at $n = 16$.
Beyond that, the density $F^{(5)}_n / T^{(5)}_n$ again decays toward
zero, though at a noticeably gentler rate: the step-to-step ratio
settles around $0.63$ rather than the $\approx\!1/2$ observed in the
unrestricted census. The restriction to minimum degree~$5$ therefore
preserves flip-symmetry slightly longer relative to the size of the
subclass, but does not alter the qualitative conclusion: even within
the minimum-degree-$5$ class --- which already contains every
candidate minimum-order $5$-chromatic graph --- flip-symmetric
examples become a vanishing fraction.
\end{document}