From 076137baaaf9e1fadd6c99a84fc95d16ae26287d Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 00:12:45 -0400 Subject: [PATCH] Add min-degree-5 flip-symmetry census through n=26 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 --- .../paper.aux | 0 .../paper.fdb_latexmk | 6 +- .../paper.fls | 2 +- .../paper.log | 6 +- .../paper.pdf | Bin 160434 -> 164862 bytes .../paper.tex | 65 +++++++++++++++--- 6 files changed, 63 insertions(+), 16 deletions(-) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.aux (100%) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.fdb_latexmk (98%) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.fls (99%) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.log (99%) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.pdf (53%) rename papers/{maximal_planar_graph_edge_flipping => flip_symmetric_maximal_planar_graphs}/paper.tex (74%) diff --git a/papers/maximal_planar_graph_edge_flipping/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux similarity index 100% rename from papers/maximal_planar_graph_edge_flipping/paper.aux rename to papers/flip_symmetric_maximal_planar_graphs/paper.aux diff --git a/papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk similarity index 98% rename from papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk rename to papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk index c9edf33..ea19a55 100644 --- a/papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778730894 "paper.tex" "paper.pdf" "paper" 1778730894 +["pdflatex"] 1778731932 "paper.tex" "paper.pdf" "paper" 1778731932 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -120,8 +120,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1778730894 1086 27c5b16223020a876dd8d169d7dbeb8e "pdflatex" - "paper.tex" 1778730888 8284 c20bb0e0a06c6c5be3373212d2684cdf "" + "paper.aux" 1778731932 1086 27c5b16223020a876dd8d169d7dbeb8e "pdflatex" + "paper.tex" 1778731732 10481 5b166fa0a035792f320327355568546e "" (generated) "paper.aux" "paper.log" diff --git a/papers/maximal_planar_graph_edge_flipping/paper.fls b/papers/flip_symmetric_maximal_planar_graphs/paper.fls similarity index 99% rename from papers/maximal_planar_graph_edge_flipping/paper.fls rename to papers/flip_symmetric_maximal_planar_graphs/paper.fls index 90deaa0..1b2b4cb 100644 --- a/papers/maximal_planar_graph_edge_flipping/paper.fls +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.fls @@ -1,4 +1,4 @@ -PWD /Users/didericis/Code/math-research/papers/maximal_planar_graph_edge_flipping +PWD /Users/didericis/Code/math-research/papers/flip_symmetric_maximal_planar_graphs INPUT /usr/local/texlive/2022/texmf.cnf INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt diff --git a/papers/maximal_planar_graph_edge_flipping/paper.log b/papers/flip_symmetric_maximal_planar_graphs/paper.log similarity index 99% rename from papers/maximal_planar_graph_edge_flipping/paper.log rename to papers/flip_symmetric_maximal_planar_graphs/paper.log index 7b2012d..25b8e89 100644 --- a/papers/maximal_planar_graph_edge_flipping/paper.log +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 13 MAY 2026 23:54 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 00:12 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -506,7 +506,7 @@ hs \OT1/cmr/m/n/10 with \OT1/cmtt/m/n/10 minimum[]connectivity $\OT1/cmr/m/n/10 Here is how much of TeX's memory you used: 13153 strings out of 478268 265523 string characters out of 5846347 - 542763 words of memory out of 5000000 + 542761 words of memory out of 5000000 30991 multiletter control sequences out of 15000+600000 475834 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 @@ -524,7 +524,7 @@ sy10.pfb> -Output written on paper.pdf (3 pages, 160434 bytes). +Output written on paper.pdf (3 pages, 164862 bytes). 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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}