even_level: extend conjecture test to the cyclically-5-connected case (n=24)

Add the n=24 result to the Even Level Graph Generators paper: the dual of
the unique 44-vertex non-Hamiltonian cyclically-5-connected cubic planar
graph (Holton-McKay Fig. 2.10) -- a 24-vertex 5-connected triangulation,
the first conjecture test outside the 3-cut family -- is a bridge-derived
level graph, two verified bridge switches from an Even Level Graph
(source 19).

- Generate the graph rather than transcribe it: plantri -c5 lists all 6833
  5-connected 24-vertex triangulations; exactly one has a non-Hamiltonian
  dual, which also settles the uniqueness Holton-McKay left open at 44
  vertices (cyclically-5-connected triangulation <=> dual cubic graph).
- New abstract sentence + "cyclically-5-connected case: n=24" subsection,
  noting the classic 46-vertex Tutte graph is only cyclically 3-connected.
- Figure 6 (figures/fig210_dual.png): the dual T, parity-coloured, with the
  two introduced bridge edges {6,19} and {20,22} in green (style of Fig. 5).
- Experiments: test_fig210_dual_bridge.py (generate->filter->test pipeline),
  verify_fig210_witness.py (step-verifies the witness), draw_fig210_dual.py
  (figure), fig210_dual.g6 (the unique graph). paper.pdf rebuilt (10 pages).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

papers/even_level_graph_generators/paper.tex

@@ -101,7 +101,12 @@ Hamiltonian, so every triangulation on at most $20$ vertices is an
 intertwining tree and the first possible failures occur at $n = 21$, at
 the six duals of the Holton--McKay graphs. We verify that all six are
 bridge-derived level graphs, confirming the conjecture in its first
-nontrivial case.
+nontrivial case. Pushing further, we identify by exhaustive generation
+the unique $44$-vertex non-Hamiltonian \emph{cyclically $5$-connected}
+cubic planar graph -- settling a uniqueness question Holton--McKay left
+open -- whose $24$-vertex $5$-connected dual is the first test of the
+conjecture outside the $3$-cut family; it too is a bridge-derived level
+graph, two bridge switches from an Even Level Graph.
 \end{abstract}
 
 \maketitle
@@ -528,6 +533,71 @@ with their Even Level Graphs and have no added edge.}
 \label{fig:n21-duals}
 \end{figure}
 
+\subsection*{The cyclically-$5$-connected case: $n = 24$}
+
+The six $n = 21$ duals all carry non-trivial $3$-cuts in the cubic
+picture; dually, each contains a separating triangle, so each is built
+from smaller pieces and lies in the most reducible part of the
+non-Hamiltonian world. (The famous $46$-vertex Tutte graph is no
+improvement here: it too is only cyclically $3$-connected, and its
+$25$-vertex dual has separating triangles.) The genuinely new regime is
+the \emph{cyclically $5$-connected} one, dual to a $5$-connected
+triangulation -- no separating $3$- or $4$-cycle, hence nothing to
+decompose along. By Holton--McKay, the smallest non-Hamiltonian
+cyclically $5$-connected cubic planar graph has $44$ vertices (Fig.~2.10
+of~\cite{holton-mckay}, attributed to Tutte; minimality due to
+Faulkner--Younger), and its dual is a $24$-vertex $5$-connected
+triangulation.
+
+We obtain this graph by generation rather than transcription. A
+$44$-vertex cubic planar graph is the dual of a $24$-vertex
+triangulation, and a cubic graph is cyclically $5$-connected if and only
+if its dual triangulation is $5$-connected. Enumerating all $5$-connected
+triangulations on $24$ vertices (\texttt{plantri -c5}, $6833$ of them)
+and testing each dual for Hamiltonicity, we find that \emph{exactly one}
+has a non-Hamiltonian dual. This both produces the graph and, granting
+the correctness of the generator and the Hamiltonicity test, settles the
+uniqueness question Holton--McKay left open: there is a unique
+non-Hamiltonian cyclically $5$-connected cubic planar graph on $44$
+vertices.
+
+Let $T$ be its dual: a $24$-vertex triangulation with vertex connectivity
+$5$ and no separating triangle, and -- since its dual is non-Hamiltonian
+-- not an intertwining tree. We find that $T$ is nonetheless a
+bridge-derived level graph. Of its $333$ valid parity partitions most are
+useless: their backward bridge-orbits exceed $8 \times 10^5$ states with
+no Even Level Graph in sight. But one partition has a backward orbit of
+only $4678$ states containing an Even Level Graph (source $s = 19$,
+maximum level $4$) at depth $2$. The two bridge switches carrying that
+Even Level Graph to $T$ are
+\[
+  \text{remove } \{16,21\},\ \text{add } \{20,22\}
+  \quad\text{and}\quad
+  \text{remove } \{15,18\},\ \text{add } \{6,19\},
+\]
+each adding a same-parity edge that is a bridge of the (odd, resp.\ even)
+parity subgraph; both steps have been verified to be valid bridge
+switches. So the disjunction holds for $T$ through the bridge-derived
+disjunct, and the ``one good partition suffices'' phenomenon seen at
+$n = 21$ persists into the cyclically $5$-connected regime -- the first
+test of the conjecture genuinely outside the $3$-cut family.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.7\textwidth]{figures/fig210_dual.png}
+\caption{The $24$-vertex dual $T$ of the unique $44$-vertex
+non-Hamiltonian cyclically $5$-connected cubic planar graph
+(Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed
+parity labelling (blue even, orange odd). $T$ is $5$-connected and not an
+intertwining tree, yet is a bridge-derived level graph: the two solid
+green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by
+the two bridge switches carrying its witness Even Level Graph (source
+$19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6,
+19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and
+in particular no odd cycle, is created.}
+\label{fig:n24-dual}
+\end{figure}
+
 \begin{thebibliography}{9}
 
 \bibitem{holton-mckay}