- 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>
Disjunction (every maximal planar graph is a derived level graph or
intertwining tree) holds through n=12. New intertwining-only iso class
at n=12 (analog of T*_9 at n=9) brings the count of derived-resistant
iso classes to 2 in this range. Per the intertwining-tree ⟺
Hamiltonian-dual equivalence, intertwining-tree failures cannot occur
until n=21 (dual of the 38-vertex Holton-McKay minimum Tait
counterexample).
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