Relocate the standalone Python scripts from the repo root into the
experiments/ folder of the paper each one belongs to:
plane_depth_sequencing/experiments/
plane_depth_sequencing.py, draw_quad_sequence.py,
draw_quad_sequence_diagram.py, extract_sequence.py,
plane_depth_sequencing_figure.py, quad_sequence_coloring_check.py
colored_edge_flip_classes/experiments/ colored_edge_flip_class_survey.py
colored_pentagon_contractions/experiments/ colored_pentagon_contractions.py
plane_diamond_coloring/experiments/ plane_diamond_coloring.py
Each file that imports lib.* (still in the repo root) or the sibling
plane_depth_sequencing module gets a sys.path shim that prepends the
repo root (computed three levels up) and, where needed, its own dir.
Imports verified to resolve from a neutral working directory.
flip_symmetric_census.py is intentionally left in the root.
Co-Authored-By: Claude Opus 4.7 (1M context) <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>
Exhaustive enumeration at order 28 finished with exactly four maximal
planar graphs of minimum degree 5 lacking a plane diamond coloring,
out of 1,204,737 total. Adds the fourth counterexample's canonical
graph6 string and updates the figure caption.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds search_min_degree_counterexample_comprehensive iterating Sage's
planar_graphs generator with minimum_degree=5. Exhaustive enumeration
through order 27 (456,967 maximal planar graphs of minimum degree at
least 5) finds no counterexample to Conjecture 2.4. At order 28, three
counterexamples are exhibited and verified via Sage's chromatic_number
on the auxiliary graph, refuting the conjecture. Updates paper with the
refutation theorem, the per-order census, a figure of one counterexample,
and graph6 strings of the other two.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Frames the paper around the scaffold-first 4-coloring program: 2-color
the BFS-layered bipartite spanning subgraph (the diamond scaffold),
then promote select vertices with two new colors to absorb the
discarded same-layer edges. Reintroduces the diamond scaffold
definition (removed in b5a9030 along with the equivalence machinery)
since it now plays a motivational rather than definitional role.
Replaces hardcoded definition/theorem/conjecture numbers with stable
\ref{}-based cross-references.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the scaffold-mediated definition with the equivalent direct
condition (two color classes contained in opposite-parity BFS layers
from some root) and removes the scaffold definition, 2-colorability
theorem, connectedness lemma, and equivalence proposition that existed
solely to translate between the two formulations. Updates the
refutation proof to invoke the new definition directly.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds search_counterexample_comprehensive iterating Sage's planar_graphs
generator across all maximal planar graphs of bounded order. Exhaustive
enumeration through order 13 (9150+49566 triangulations) yields exactly
one graph with no plane diamond coloring, at order 13. Updates Theorem
2.6 to assert minimality and uniqueness, and replaces the figure and
edge list with the smaller counterexample.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds Theorem 2.6 stating Conjecture 2.5 is false, with proof exhibiting
a 16-vertex maximal planar graph (graph6 O???IAGKEBEQQYHdplW{n) for
which the auxiliary 4-colorability check fails at every root vertex,
verified computationally via Sage's chromatic_number. Includes the
graph as a figure and adds a McKay graph6 reference.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Extends paper with: a notation section for color-class preimages; the
plane diamond coloring definition (4-coloring whose two classes lift to a
2-coloring of some BFS-rooted diamond scaffold); a connectedness lemma
for the scaffold; a proposition reformulating the property as parity-
separation of two color classes by BFS layers; a remark noting this is
strictly stronger than 4CT; and the conjecture that every maximal planar
graph admits such a coloring.
Adds plane_diamond_coloring.py with get_plane_diamond_scaffold and a
counterexample search that reduces the per-root check to 4-colorability
of an auxiliary graph forcing two colors onto opposite parity layers.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Defines distance partition (BFS layers from a chosen vertex) and the diamond
scaffold of a maximal planar graph (G with all same-level edges removed),
then proves the diamond scaffold is 2-colorable by parity of level.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis