Enumerate non-Hamiltonian cyclically-5-connected cubic planar graphs by
running plantri -c5 -d for n in {23,25,26} (n=24 already in the previous
commit) and filtering for non-Hamiltonian dual:
n=23 -> 0 of 1970 (recomputes Faulkner-Younger minimality)
n=24 -> 1 of 6833 (the Tutte/Fig 2.10 graph)
n=25 -> 1 of 23384 (new; unique 46-vertex one)
n=26 -> 0 of 82625
Both T (n=24) and T_25 (n=25) verified internally 6-connected by exhaustive
5-cut scan: every 5-cut is the neighborhood of a degree-5 vertex. This is
the strongest connectivity a planar triangulation can have and the level
at which Birkhoff-style reductions terminate, so both are genuinely
irreducible bases of any decomposition argument.
T_25 is also bridge-derived: witness Even Level Graph from source 24
(max level 4) at depth 2, orbit only 3114 states. Forward switches:
remove {21,23} add {22,24}; remove {3,5} add {1,6}. Both adds are bridges
of the even parity subgraph. Same witness signature as T (minimum total
Betti, tiny orbit, depth 2).
New subsection "Beyond n=24: enumeration and the next 5-connected core",
abstract extended, new Figure 7 (core_n25_dual.png). Reproducibility
scripts: draw_core_witness.py and verify_core_witness.py (both
parametrized so they work on any 5-conn non-Ham-dual core's g6).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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>
Remove the witness-ELG figure (former Fig. 5); keep the six resulting duals
with their introduced green bridge edges. Fix the dangling cross-reference
in the caption.
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>
Figure fig:n21-witnesses: each of the six Holton-McKay duals drawn as its
witness Even Level Graph in a radial-by-level layout (source centre,
level-k vertices on ring k), coloured by parity. Dashed red edges are the
flipped same-parity edges and solid green edges the introduced bridges;
applying the switches yields the dual. Duals 1,2 are ELGs outright.
draw_witnesses.py generates the combined 2x3 figure and per-dual PNGs from
the verified witness JSONs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis