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>
Record the partition sweep on the n=24 Fig 2.10 dual. New subsection +
experiments/bridge_partition_sweep.py.
Findings:
- A bridge switch is a constrained diagonal flip; bridge-derived via L
means lying in an Even-Level-Graph component of the restricted flip
graph. So the question is which flip-components contain an ELG.
- Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each
face has one within-pair edge), so total parity-subgraph Betti =
(c_A+c_B)-2; intertwining trees are the Betti-0 case.
- Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3;
min is 1 (T not intertwining). All 27 partitions found bridge-derived
(depth 2-3) have the minimum Betti 1 -> necessary.
- But not sufficient: only 27 of 288 Betti-1 partitions yield a witness;
the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase
found none. The discriminator is flip-component structure (sharp
orbit-size dichotomy), not a numerical invariant. Characterizing which
Betti-minimal partitions sit in an ELG component is left open.
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>
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>
- Add Theorem: maximal planar G is an intertwining tree iff its dual
G* is Hamiltonian (Tait-style Jordan-curve argument). Consequence:
smallest non-intertwining-tree triangulations are the 6 duals of the
38-vertex Holton-McKay graphs, at n=21.
- Load the 6 graphs from McKay's authoritative planar_code file
(nonham38m4.pc), verified: 38 vertices, cubic, planar, non-Hamiltonian.
- All 6 duals confirmed not intertwining trees (exhaustive 2^20 check).
- 2 of 6 duals are themselves Even Level Graphs (sources 9, 10), hence
derived level graphs -- first cases where the derived disjunct does
work the intertwining-tree disjunct cannot.
- Remaining 4: bounded E/O-orbit search inconclusive; status open. This
is the first genuinely undetermined instance of the conjecture.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- New paper papers/even_level_graph_generators/: defines Even Level
Graph (every level cycle even), derived level graphs, intertwining
trees, and the disjunction conjecture (every maximal planar graph is
a derived level graph or intertwining tree). Empirically tested
through n=11: every iso class is at least an intertwining tree, so
the disjunction holds trivially in this range. The intertwining tree
disjunct fails at the Tutte graph dual (n=25), so the disjunction
becomes non-trivial past some unknown threshold.
- Level Switching paper: adds Section 4 (Reachability via edge
switches) with the two-step argument (Sleator-Tarjan-Thurston for
Case 1; face-merges for Case 2) and Theorem 4.1 (O(n) edge switches
suffice to reach all-depth-0).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis