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>
Records, for 4<=n<=11, triangulation iso classes, how many admit an ELG
source, ELG iso classes, and the automorphism-free flag-rooted count
sum_G 4E/|Aut(G)| * s(G). Computed by experiments/count_elgs.py.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
experiments/test_tutte_bridge.py: bridge-derivability test for the dual of
the 46-vertex Tutte graph (a 25-vertex non-intertwining triangulation,
since the Tutte graph is non-Hamiltonian) -- the conjecture's first case
beyond the n=21 Holton-McKay duals. Reuses the fast integer-state bridge
engine: per source labelling with bipartite parity subgraphs, run a
backward bridge-orbit BFS for an Even Level Graph witness.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replace the radial (crossing-heavy) figure with two crossing-free planar
drawings (networkx planar_layout / Chrobak-Payne):
fig:n21-elgs -- the six witness Even Level Graphs, parity-coloured, with
the bridge-switch-flipped edges dashed red;
fig:n21-duals -- the six resulting duals, with the introduced bridge edges
solid green.
ELG and dual are drawn with independent planar layouts so neither has any
edge crossing (a flip diagonal would otherwise cross other edges when its
quadrilateral is non-convex, which happens for duals 0 and 3). Drop forced
equal aspect so panels fill and labels separate.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Tested duals 1 and 2: both are Even Level Graphs directly (dual 1 for
source 10, dual 2 for source 9), so bridge-derived with a zero-length
switch sequence. All six Holton-McKay duals are confirmed non-intertwining
(consistent with the dual-Hamiltonian theorem, since all six HM graphs are
non-Hamiltonian) and all six are bridge-derived. Saved witness files
dual_1.json, dual_2.json (0 switches) to complete the archive for all six.
Updated the n=21 subsection accordingly.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis