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