Retitle to "Even Level Graph Generators: a constructive conjecture
stronger than the Four Color Theorem" and state explicitly in the
abstract and introduction that the conjecture implies the four color
theorem but is strictly stronger: a 4-coloring grouped {1,2}|{3,4} is
exactly a partition into two bipartite-inducing parts, so 4CT is the bare
existence of such a partition, whereas the conjecture demands it be
realized constructively (bridge-switch level parity, or two induced
trees). Hence a proof is a new constructive proof of 4CT, and the
conjecture is at least as hard -- very likely harder.
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>
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>
Frame the paper's purpose: ask whether two constructive families of
4-colorable triangulations -- bridge-derived level graphs (parity
2-coloring) and intertwining trees (two trees, disjoint color pairs) --
suffice to generate every maximal planar graph on n vertices. An
affirmative answer would be a constructive proof of the four color theorem
for triangulations. State the duality bridge to Tait/Holton-McKay and the
n=21 confirmation.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Drop the n<=9 bridge-derived classification table and its surrounding
discussion; the n=21 boundary case now follows directly from the
trivial-below-21 observation.
Co-Authored-By: Claude Opus 4.7 <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>
Table tab:n21 records, for each of the six duals: not an intertwining tree;
Even Level Graph source (duals 1,2 only); and bridge-switch path length to
an ELG (0,0 for the two ELG-outright cases; 3,1,2,4 for the rest). All six
are bridge-derived; all witnesses step-verified.
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>
- Conjecture now reads "bridge-derived level graph ... an intertwining tree,
or both" -- the stronger form the evidence actually supports (a bridge-
derived level graph is automatically a valid derived level graph).
- Empirical table recomputed for bridge-derivability, exhaustively for n<=9
(every backward bridge-orbit fully enumerable there):
n=7: 1 inter-only; n=8: 2 inter-only; n=9: 14 inter-only; missing=0.
Added prose: below n=21 every class is intertwining, so the table shows
how far the bridge-derived disjunct reaches on its own (36/50 at n=9) and
that the two disjuncts complement each other; "bridge only" is 0 in range.
- n=21 subsection notes the four witnesses are explicit, short (path lengths
3,1,2,4), archived, and step-verified.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
- 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