Backward bridge-switch search (sharded over valid parity partitions) found
an Even Level Graph witness for each of the four previously-open duals:
dual 0: partition 12, witness orbit 9458
dual 3: partition 9, witness orbit 388
dual 4: partition 23, witness orbit 3842
dual 5: partition 12, witness orbit 165668
So all four are bridge-derived level graphs, hence valid derived level
graphs. Combined with the two duals that are Even Level Graphs outright,
the disjunction is now confirmed for ALL SIX critical iso classes at n=21
-- the first nontrivial test of the conjecture passes.
Why it worked where exhaustion failed: a witness, when it exists, tends to
sit in a SMALL orbit (here a few hundred to ~1.7e5 states) reachable
quickly, while other parity partitions of the same triangulation have
orbits >1e6. We only need one good partition. The bridge restriction both
shrinks orbits ~100x and guarantees validity, so any ELG found in a
backward orbit is an immediate witness.
- Update paper n=21 subsection to report the resolution.
- Add shard_hunt.py (partition-sharded parallel witness hunt).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Findings at n=9 (50 triangulations, orbits fully exhaustible):
- 36 bridge-derived, 14 NOT bridge-derived. So bridge-derived is a PROPER
subclass of derived (49 derived at n=9). All 14 non-bridge graphs are
intertwining trees -- as are all 50, necessarily: intertwining tree
<=> dual Hamiltonian, and the smallest non-Hamiltonian 3-connected cubic
planar graph has 38 vertices, i.e. dual on 2n-4=38 => n=21. Hence every
triangulation with n<=20 is an intertwining tree, and the disjunction
"bridge-derived OR intertwining" is trivially true below n=21. The 4
Holton-McKay duals are the first non-intertwining triangulations.
- Static parity-subgraph invariants (Betti numbers, component counts,
cross-edge count, existence of an all-forest partition) do NOT separate
bridge-derived from non-bridge-derived -- both classes realize beta=0
partitions and identical ranges. Bridge-derivability is dynamical, not a
simple static invariant; no easy obstruction.
- Side lemma: every valid parity partition of an n-vertex triangulation has
exactly 2n-4 cross edges (intra-edges = n-2). Holds for all n=9 graphs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- fast_bridge.py: states as 210-bit integer edge-bitmasks (compact memory,
O(1) set ops); build a NetworkX graph only once per state for the planar
embedding; parity-subgraph bridges via one iterative DFS per state instead
of per-edge subgraph copies. Validated identical orbits to the slow version;
throughput ~5170 states/s vs ~1100 (graph.copy was 66% of old runtime).
- fast_decide.py: integrated, gated ELG-witness check (only even-class
sources with all-opposite-class neighbourhoods are tested with the
ground-truth is_even_level_graph, then parity match). Witness detection
validated (ELGs -> True, T*_9 -> False).
- Feasibility finding: bridge orbits are ~100x smaller than full E/O orbits
but still 1e5-1e6 states per labelling (partitions 0,1 of dual 0 exceed
310k and 685k without exhausting), x ~150 valid parity partitions per dual.
Exhausting every orbit -- required for a conclusive NEGATIVE -- is
computationally infeasible. A conclusive POSITIVE (witness ELG) remains
reachable; none found so far.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Define bridge switch (E/O switch whose new same-parity edge is a bridge
in its parity subgraph) and bridge-derived level graph in the paper.
Note that bridge switches preserve bipartite parity subgraphs, so every
bridge-derived level graph is automatically valid.
- Discover the E/O-switch relation is directed (irreversible when a switch
produces a cross-parity edge); T*_9 reaches an ELG forward but no ELG
reaches it, explaining why it is not derived. This rules out a simple
switch-invariant characterization.
- Bridge orbits are far smaller than full E/O orbits (~10^4 vs ~10^8 for
some labellings), making exhaustive search feasible. Each of the 4 open
duals has ~150 valid parity partitions; exhaustive bridge-orbit search
per partition can decide bridge-derivability conclusively.
Co-Authored-By: Claude Opus 4.7 <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