diff --git a/flip_symmetric_census.py b/flip_symmetric_census.py new file mode 100644 index 0000000..d09c04e --- /dev/null +++ b/flip_symmetric_census.py @@ -0,0 +1,73 @@ +"""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) diff --git a/papers/maximal_planar_graph_edge_flipping/paper.aux b/papers/maximal_planar_graph_edge_flipping/paper.aux new file mode 100644 index 0000000..b17b3d3 --- /dev/null +++ b/papers/maximal_planar_graph_edge_flipping/paper.aux @@ -0,0 +1,16 @@ +\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} +\newlabel{tocindent3}{0pt} +\gdef \@abspage@last{3} diff --git a/papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk b/papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk new file mode 100644 index 0000000..c9edf33 --- /dev/null +++ b/papers/maximal_planar_graph_edge_flipping/paper.fdb_latexmk @@ -0,0 +1,128 @@ +# Fdb version 3 +["pdflatex"] 1778730894 "paper.tex" "paper.pdf" "paper" 1778730894 + "/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 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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} + +%----------------------------------------------------------------------- +% End of amsart-template.tex +%-----------------------------------------------------------------------