Add self-flip-neighbor survey and write-up

Add experiments/self_flip_neighbor.py: enumerate the admissible flips of a
maximal planar graph and test each for isomorphism to the original, i.e. decide
whether G lies in its own flip neighborhood N(G). Supports a single graph6 input
or an order-range survey over min-degree-5 triangulations.

Add a "self-flip neighbors" section to paper.tex with the n=12..25 survey table
and a remark: the self-flip fraction declines as expected (min-degree-5
triangulations are asymptotically rigid), and this is not useful for narrowing
the minimal criminal since it neither includes nor excludes any candidate.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/colored_edge_flip_classes/experiments/self_flip_neighbor.py

@@ -0,0 +1,103 @@
+"""Experiment: is a given maximal planar graph in its own flip neighborhood?
+
+The flip neighborhood N(G) of a maximal planar graph G (Definition
+"flip-neighborhood" in paper.tex) is the set of maximal planar graphs obtainable
+from G by a single admissible edge flip: pick an edge uv whose two incident
+triangular faces are uvw and uvx, delete uv, and insert wx, provided wx is not
+already an edge.
+
+We call G a *self-flip-neighbor* iff some admissible flip G^flip(uv) is
+isomorphic to G --- equivalently, G in N(G). A flip always changes the labelled
+edge set (it removes uv and adds a non-edge wx), so this is a genuine question
+about the isomorphism type: it asks whether a single diagonal flip can map the
+triangulation back onto a copy of itself.
+
+This module enumerates every admissible flip of G, tests each resulting
+triangulation for isomorphism to G, and reports the witnessing edges. It can run
+on a single graph (given as a graph6 string, or the icosahedron by default) or
+survey every min-degree-5 maximal planar graph over a range of orders.
+"""
+import os, sys
+sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), "..", "..", "..")))  # repo root for `lib`
+from typing import Iterator, Any, cast
+from sage.all import Graph, graphs  # type: ignore[attr-defined]  # pylint: disable=no-name-in-module
+
+# Reuse the flip machinery from the survey in this same directory.
+from colored_edge_flip_class_survey import face_thirds, canonical_g6  # type: ignore[import]
+
+
+def admissible_flips(g: Graph) -> Iterator[tuple[Any, Any, Any, Any, Graph]]:
+    """Yield (u, v, w, x, H) for every admissible edge flip of g: uv is the
+    flipped edge, wx the inserted diagonal (a non-edge), and H = g^flip(uv)."""
+    pairs = face_thirds(g)
+    for u, v in list(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
+        h = g.copy()
+        h.delete_edge(u, v)
+        h.add_edge(w, x)
+        yield u, v, w, x, h
+
+
+def self_flip_witnesses(g: Graph) -> list[tuple[Any, Any, Any, Any]]:
+    """Return every (u, v, w, x) such that the admissible flip uv -> wx yields a
+    graph isomorphic to g. Non-empty iff g lies in its own flip neighborhood."""
+    target = canonical_g6(g)
+    witnesses: list[tuple[Any, Any, Any, Any]] = []
+    for u, v, w, x, h in admissible_flips(g):
+        if canonical_g6(h) == target:
+            witnesses.append((u, v, w, x))
+    return witnesses
+
+
+def is_self_flip_neighbor(g: Graph) -> bool:
+    """True iff g in N(g): some admissible flip of g is isomorphic to g."""
+    target = canonical_g6(g)
+    return any(canonical_g6(h) == target for *_e, h in admissible_flips(g))
+
+
+def report_single(g: Graph) -> bool:
+    """Print a per-edge flip report for g and return whether it self-flips."""
+    flips = list(admissible_flips(g))
+    witnesses = self_flip_witnesses(g)
+    print(f"graph6={g.graph6_string()}  |V|={g.order()}  |E|={g.size()}  "
+          f"admissible flips={len(flips)}  self-flip witnesses={len(witnesses)}")
+    for u, v, w, x in witnesses:
+        print(f"    flip edge ({u},{v}) -> ({w},{x}) gives a graph isomorphic to G")
+    print(f"  => G {'IS' if witnesses else 'is NOT'} in its own flip neighborhood")
+    return bool(witnesses)
+
+
+def survey(min_order: int, max_order: int) -> None:
+    """For each order in [min_order, max_order], count how many min-degree-5
+    maximal planar graphs are self-flip-neighbors."""
+    for n in range(min_order, max_order + 1):
+        gen = graphs.planar_graphs(
+            n, minimum_connectivity=3, maximum_face_size=3, minimum_degree=5
+        )
+        checked = 0
+        self_flip = 0
+        for g in gen:
+            checked += 1
+            if is_self_flip_neighbor(cast(Graph, g)):
+                self_flip += 1
+        print(f"order {n}: {checked} min-degree-5 maximal planar graphs, "
+              f"{self_flip} are self-flip-neighbors")
+
+
+if __name__ == "__main__":
+    if len(sys.argv) >= 2 and sys.argv[1].lstrip("-").isdigit():
+        lo = int(sys.argv[1])
+        hi = int(sys.argv[2]) if len(sys.argv) > 2 else lo
+        survey(lo, hi)
+    else:
+        if len(sys.argv) >= 2:
+            graph = Graph(sys.argv[1])
+        else:
+            graph = graphs.IcosahedralGraph()
+            print("(no graph given; using the icosahedron)")
+        report_single(cast(Graph, graph))

papers/colored_edge_flip_classes/paper.aux

@@ -10,12 +10,16 @@
 \newlabel{lem:edge-deletion-4colorable}{{4.2}{2}}
 \newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}}
 \newlabel{thm:flip-neighborhood-4colorable}{{4.4}{3}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Case\nonbreakingspace 2 of the proof of Theorem\nonbreakingspace 4.4\hbox {}: $u, v$ share color $a$ and $w, x$ share color $c$. The $\{a, b\}$-Kempe path $P$ from $u$ to $v$ separates $w$ from $x$ in the plane, so no $\{c, d\}$-path between $w$ and $x$ can avoid crossing $P$; since the color sets $\{a, b\}$ and $\{c, d\}$ are disjoint, no such path exists.}}{4}{}\protected@file@percent }
+\newlabel{fig:flip-proof-case-two}{{2}{4}}
+\newlabel{thm:no-colored-class-contains-G}{{4.5}{4}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A computational aside: self-flip neighbors}}{4}{}\protected@file@percent }
+\newlabel{def:self-flip-neighbor}{{5.1}{4}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{0pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Case\nonbreakingspace 2 of the proof of Theorem\nonbreakingspace 4.4\hbox {}: $u, v$ share color $a$ and $w, x$ share color $c$. The $\{a, b\}$-Kempe path $P$ from $u$ to $v$ separates $w$ from $x$ in the plane, so no $\{c, d\}$-path between $w$ and $x$ can avoid crossing $P$; since the color sets $\{a, b\}$ and $\{c, d\}$ are disjoint, no such path exists.}}{4}{}\protected@file@percent }
+\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Self-flip neighbors among min-degree-$5$ maximal planar graphs. No graph on $n \leq 15$ vertices is a self-flip neighbor; beyond the first occurrences the fraction declines steadily.}}{5}{}\protected@file@percent }
-\newlabel{fig:flip-proof-case-two}{{2}{4}}
+\newlabel{tab:self-flip}{{1}{5}}
-\newlabel{thm:no-colored-class-contains-G}{{4.5}{4}}
+\gdef \@abspage@last{5}
-\gdef \@abspage@last{4}

papers/colored_edge_flip_classes/paper.fdb_latexmk

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papers/colored_edge_flip_classes/paper.tex

@@ -371,6 +371,79 @@ in particular, $\varphi$ is a proper $4$-coloring of $H_k = G$.  But
 $\chi(G) \geq 5$ admits no such coloring, a contradiction.
 \end{proof}
 
+\section{A computational aside: self-flip neighbors}
+
+Since $\mathcal{N}(G)$ is defined up to isomorphism, it is natural to
+ask whether $G$ can be one of its own flip neighbors.
+
+\begin{definition}[Self-flip neighbor]\label{def:self-flip-neighbor}
+A maximal planar graph $G$ is a \emph{self-flip neighbor} if
+$G \in \mathcal{N}(G)$; that is, if some admissible edge flip
+$G^{\mathrm{flip}(uv)}$ is isomorphic to $G$.
+\end{definition}
+
+A single flip always changes the labelled edge set --- it deletes
+$uv$ and inserts a non-edge $wx$ --- so the property is genuinely one
+of the isomorphism type: it asks whether a diagonal flip can carry the
+triangulation back onto a copy of itself.  Equivalently, the degree
+changes induced by the flip ($-1$ at $u$ and $v$, $+1$ at $w$ and $x$)
+must be undone by an automorphism of the resulting graph.
+
+We surveyed every maximal planar graph of minimum degree $5$ on
+$n \leq 25$ vertices, counting those that are self-flip neighbors.
+The results are recorded in Table~\ref{tab:self-flip}.
+
+\begin{table}[h]
+\centering
+\begin{tabular}{rrrr}
+\hline
+$n$ & min-degree-$5$ & self-flip & fraction \\
+    & triangulations & neighbors  &           \\
+\hline
+$12$ & $1$     & $0$   & $0\%$    \\
+$13$ & $0$     & $0$   & ---      \\
+$14$ & $1$     & $0$   & $0\%$    \\
+$15$ & $1$     & $0$   & $0\%$    \\
+$16$ & $3$     & $1$   & $33.3\%$ \\
+$17$ & $4$     & $1$   & $25.0\%$ \\
+$18$ & $12$    & $2$   & $16.7\%$ \\
+$19$ & $23$    & $5$   & $21.7\%$ \\
+$20$ & $73$    & $12$  & $16.4\%$ \\
+$21$ & $192$   & $27$  & $14.1\%$ \\
+$22$ & $651$   & $51$  & $7.8\%$  \\
+$23$ & $2070$  & $120$ & $5.8\%$  \\
+$24$ & $7290$  & $273$ & $3.7\%$  \\
+$25$ & $25381$ & $598$ & $2.4\%$  \\
+\hline
+\end{tabular}
+\caption{Self-flip neighbors among min-degree-$5$ maximal planar
+graphs.  No graph on $n \leq 15$ vertices is a self-flip neighbor;
+beyond the first occurrences the fraction declines steadily.}
+\label{tab:self-flip}
+\end{table}
+
+\begin{remark}
+The proportion of self-flip neighbors declines as $n$ grows --- from a
+peak near a third at $n = 16$ to roughly $2\%$ at $n = 25$ --- and the
+trend is the one to expect.  A self-flip neighbor requires a flip whose
+induced degree changes are reversed by an automorphism of the flipped
+graph, but min-degree-$5$ triangulations are asymptotically rigid: the
+share with a nontrivial automorphism group, let alone one of the
+required form, tends to $0$.  The absolute count of self-flip neighbors
+continues to grow, but ever more slowly than the census itself, so the
+fraction appears to vanish in the limit.
+
+This observation is not, by itself, useful for narrowing down the
+minimal criminal $G_0$.  Whether or not $G_0$ is a self-flip neighbor,
+Theorem~\ref{thm:flip-neighborhood-4colorable} already shows every
+graph in $\mathcal{N}(G_0)$ is $4$-colorable; self-flip neighborness
+is a generic structural feature of the surrounding census rather than a
+property forced on, or forbidden of, a minimum counterexample.  It
+neither includes nor excludes any candidate, and so contributes nothing
+to the elimination program beyond confirming that the flip operation
+behaves on min-degree-$5$ triangulations as one would anticipate.
+\end{remark}
+
 \end{document}
 
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