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Add flip-symmetry paper with empirical density census through n=14

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Introduces the maximal_planar_graph_edge_flipping paper, motivating
flip-symmetry as a structural restriction on minimum-order
five-chromatic counterexamples, and reports an exhaustive census
showing |F_n|/|T_n| decays geometrically (factor ~1/2 per step from
n=10 to n=14). The census driver lives in flip_symmetric_census.py.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-13 23:56:38 -04:00
parent 256cbbf90a
commit 1749f702cf
7 changed files with 1436 additions and 0 deletions
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flip_symmetric_census.py
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"""Census of flip-symmetric maximal planar graphs.
A maximal planar graph G is *flip-symmetric* if some admissible edge uv
satisfies G^flip(uv) ~= G, where flip(uv) replaces uv with wx (the third
vertices of the two triangular faces incident to uv), and admissible
means wx is not already an edge of G.
"""
from typing import Iterator
from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module
def triangular_face_pairs(g: Graph) -> dict[frozenset, tuple]:
"""For each edge of g (a triangulation), return the two third-vertices
of its incident triangular faces, keyed by frozenset({u, v})."""
g_emb = g.copy()
if not g_emb.is_planar(set_embedding=True):
raise ValueError("graph is not planar")
pairs: dict[frozenset, list] = {}
for face in g_emb.faces():
if len(face) != 3:
raise ValueError("graph is not a triangulation")
a, b, c = face[0][0], face[1][0], face[2][0]
for u, v, third in [(a, b, c), (b, c, a), (c, a, b)]:
pairs.setdefault(frozenset((u, v)), []).append(third)
return {k: tuple(v) for k, v in pairs.items()}
def is_flip_symmetric(g: Graph) -> bool:
"""Return True iff g admits some admissible flip uv -> wx with G^flip ~= G."""
pairs = triangular_face_pairs(g)
for u, v in g.edges(labels=False):
thirds = pairs.get(frozenset((u, v)))
if thirds is None or len(thirds) != 2:
continue
w, x = thirds
if g.has_edge(w, x):
continue
flipped = g.copy()
flipped.delete_edge(u, v)
flipped.add_edge(w, x)
if g.is_isomorphic(flipped):
return True
return False
def census(min_order: int, max_order: int, minimum_degree: int | None = None) -> None:
"""Enumerate every maximal planar graph of order in [min_order, max_order]
(optionally restricted by minimum_degree) and print counts and density
of flip-symmetric graphs at each order."""
for n in range(min_order, max_order + 1):
kwargs = {"minimum_connectivity": 3, "maximum_face_size": 3}
if minimum_degree is not None:
kwargs["minimum_degree"] = minimum_degree
total = 0
flip_sym = 0
gen: Iterator[Graph] = graphs.planar_graphs(n, **kwargs)
for g in gen:
total += 1
if is_flip_symmetric(g):
flip_sym += 1
if total % 1000 == 0:
print(f" order {n}: {total} checked, {flip_sym} flip-symmetric")
density = (flip_sym / total) if total else 0.0
tag = f" (min_degree {minimum_degree})" if minimum_degree is not None else ""
print(f"n={n}{tag}: total={total} flip_symmetric={flip_sym} density={density:.6f}")
if __name__ == "__main__":
import sys
max_order = int(sys.argv[1]) if len(sys.argv) > 1 else 12
min_order = int(sys.argv[2]) if len(sys.argv) > 2 else 4
min_deg = int(sys.argv[3]) if len(sys.argv) > 3 else None
census(min_order, max_order, minimum_degree=min_deg)
papers/maximal_planar_graph_edge_flipping/paper.aux
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\relax
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Preliminaries}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip-symmetric maximal planar graphs}}{1}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces An edge flip replaces the diagonal $uv$ of the quadrilateral $uwvx$ with the diagonal $wx$.}}{2}{}\protected@file@percent }
\newlabel{def:flip-symmetric}{{3.1}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{A minimal four-colorable counterexample}}{2}{}\protected@file@percent }
\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.1}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Flip symmetry frequency}}{2}{}\protected@file@percent }
\newlabel{sec:frequency}{{5}{2}}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
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\gdef \@abspage@last{3}
papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk
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%% filename: amsart-template.tex
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%% Society.
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%% ====================================================================
% AMS-LaTeX v.2 template for use with amsart
%
% Remove any commented or uncommented macros you do not use.
\documentclass{amsart}
\usepackage{tikz}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
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\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{Maximal Planar Graph Edge Flipping}
% Remove any unused author tags.
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{}
\date{}
\dedicatory{}
\begin{abstract}
\end{abstract}
\maketitle
\section{Motivation}
The Four Color Theorem asserts that every planar graph is properly
$4$-colorable, or equivalently that no maximal planar graph $G$
satisfies $\chi(G) \geq 5$. Suppose, towards a contradiction, that
such a graph exists; let $G_0$ be one of minimum order. Any structural
property shared by every maximal planar graph $H$ with $|V(H)| =
|V(G_0)|$ is then automatically inherited by $G_0$, and any property
\emph{not} satisfied by $G_0$ excludes a portion of the class of
maximal planar graphs from playing the role of a minimum
counterexample.
This paper investigates one such property: invariance under an
admissible edge flip. We call a maximal planar graph $G$
\emph{flip-symmetric} when some admissible flip at an edge of $G$
returns a graph isomorphic to $G$. Our principal observation
(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that a
minimum-order $5$-chromatic maximal planar graph cannot be
flip-symmetric, so the search for a counterexample to the Four Color
Theorem may, in principle, be confined to the complement of the class
$\mathcal{F}$ of flip-symmetric graphs. This raises a quantitative
question --- how large is $\mathcal{F}$? --- which we address
empirically in Section~\ref{sec:frequency} by an exhaustive census of
maximal planar graphs of small order.
\section{Preliminaries}
Let $G$ be a maximal planar graph with $|V(G)| \geq 4$, embedded in the
plane so that every face --- including the outer face --- is a triangle.
Every edge $uv \in E(G)$ is then shared by exactly two triangular faces
$uvw$ and $uvx$ whose union is a quadrilateral $uwvx$ with diagonal $uv$.
\begin{definition}[Edge flip]
Let $G$ be a maximal planar graph and let $uv \in E(G)$ be an edge whose
two incident triangular faces are $uvw$ and $uvx$. The \emph{edge flip}
(or \emph{diagonal flip}) at $uv$ is the operation that deletes the edge
$uv$ and inserts the edge $wx$ in its place, replacing the two triangles
$uvw$ and $uvx$ by the two triangles $uwx$ and $vwx$. The flip is
\emph{admissible} if $wx \notin E(G)$; otherwise the resulting multigraph
is not simple and the flip is forbidden.
\end{definition}
\begin{figure}[h]
\centering
\begin{tikzpicture}[
every node/.style={circle, fill=black, inner sep=1.5pt},
label distance=2pt,
scale=1.2
]
% --- before flip ---
\begin{scope}[xshift=0cm]
\node[label=left:$u$] (u) at (0,0) {};
\node[label=right:$v$] (v) at (2,0) {};
\node[label=above:$w$] (w) at (1,1) {};
\node[label=below:$x$] (x) at (1,-1) {};
\draw (u) -- (w) -- (v) -- (x) -- (u);
\draw[very thick] (u) -- (v);
\node[draw=none, fill=none] at (1,-1.6) {before};
\end{scope}
% --- arrow ---
\draw[->, very thick, shorten >=2pt, shorten <=2pt]
(2.6,0) -- node[draw=none, fill=none, above]{flip $uv$} (3.8,0);
% --- after flip ---
\begin{scope}[xshift=4.4cm]
\node[label=left:$u$] (u2) at (0,0) {};
\node[label=right:$v$] (v2) at (2,0) {};
\node[label=above:$w$] (w2) at (1,1) {};
\node[label=below:$x$] (x2) at (1,-1) {};
\draw (u2) -- (w2) -- (v2) -- (x2) -- (u2);
\draw[very thick] (w2) -- (x2);
\node[draw=none, fill=none] at (1,-1.6) {after};
\end{scope}
\end{tikzpicture}
\caption{An edge flip replaces the diagonal $uv$ of the quadrilateral
$uwvx$ with the diagonal $wx$.}
\end{figure}
\section{Flip-symmetric maximal planar graphs}
For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$
with incident triangles $uvw$, $uvx$, write
\[
G^{\mathrm{flip}(uv)} \;=\; \bigl(V(G),\,(E(G)\setminus\{uv\})\cup\{wx\}\bigr)
\]
for the graph obtained from $G$ by flipping $uv$.
\begin{definition}[Flip-symmetric graph]\label{def:flip-symmetric}
A maximal planar graph $G$ is \emph{flip-symmetric} if there exists an
admissible edge $uv \in E(G)$ such that
$G^{\mathrm{flip}(uv)} \cong G$. We write $\mathcal{F}$ for the class
of flip-symmetric maximal planar graphs.
\end{definition}
\section{A minimal four-colorable counterexample}
\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric}
Let $G$ be a maximal planar graph of minimum order among all maximal
planar graphs $H$ with $\chi(H) \geq 5$. Then $G \notin \mathcal{F}$;
that is, $G$ is not flip-symmetric.
\end{theorem}
\section{Flip symmetry frequency}\label{sec:frequency}
To gauge how restrictive flip-symmetry is, we performed an exhaustive
census of maximal planar graphs of small order. For each
$n \in \{4, 5, \dots, 14\}$ we enumerated every isomorphism class of
maximal planar graph on $n$ vertices using \texttt{plantri} (invoked
through SageMath as \texttt{graphs.planar\_graphs} with
\texttt{minimum\_connectivity}~$=3$ and
\texttt{maximum\_face\_size}~$=3$), and for each such $G$ we tested
every admissible edge $uv \in E(G)$ for the existence of an isomorphism
$G \cong G^{\mathrm{flip}(uv)}$. Writing $T_n$ for the total number of
maximal planar graphs on $n$ vertices and
$F_n = |\mathcal{F} \cap \{G : |V(G)| = n\}|$ for the number of
flip-symmetric ones, the results are tabulated below.
\begin{center}
\begin{tabular}{r r r l}
\hline
$n$ & $T_n$ & $F_n$ & $F_n / T_n$ \\
\hline
$4$ & $1$ & $0$ & $0.000000$ \\
$5$ & $1$ & $1$ & $1.000000$ \\
$6$ & $2$ & $1$ & $0.500000$ \\
$7$ & $5$ & $1$ & $0.200000$ \\
$8$ & $14$ & $5$ & $0.357143$ \\
$9$ & $50$ & $17$ & $0.340000$ \\
$10$ & $233$ & $48$ & $0.206009$ \\
$11$ & $1{,}249$ & $164$ & $0.131305$ \\
$12$ & $7{,}595$ & $552$ & $0.072679$ \\
$13$ & $49{,}566$ & $1{,}828$ & $0.036880$ \\
$14$ & $339{,}722$& $6{,}164$ & $0.018144$ \\
\hline
\end{tabular}
\end{center}
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.
\end{document}
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