Definition 1.1 (Level source) is broadened: a level source is now a set
that is either a single vertex or a simple cycle, splitting the old
notion into 'vertex source' and 'cycle source'. Downstream theorems
(Prop 1.7, Lemma 1.8, Thm 1.17) remain stated for vertex sources but
are referenced by the new material with cycle sources.
New Theorem 1.19 (Tire-tree decomposition): for any tread T in
T(G, {v_0}) at depth d >= 1 with outer cycle C_T, the sub-graph G_T
inside C_T on the side away from v_0 is a triangulated disk; taking
C_T as a cycle source, T(G_T, C_T) is canonically iso to the
sub-tree of T(G, {v_0}) rooted at T. Proof in three steps:
(D1) triangulated-disk via Jordan curve, (D2) level-shift
ell_{G_T}(.) = ell_G(.) - d via shortest-path stays in R_T, (D3)
component-of-G'_k bijection with descendants of T.
Figure fig_tire_tree_decomposition.png (and its generator
experiments/draw_tire_tree_decomposition.py) illustrates the
decomposition on a 13-vertex, 5-level example with four nested seams
C_{T_R}, C_{T_L}, C_{T_{LL}}, C_{T_{LLL}}; the generator script
verifies the level-shift assertion on this instance. Vertex
positions are hand-tuned in TikZiT and copied back; the right-panel
labels are rotated relative to the parent G to emphasise the new
role of C_{T_L} as cycle source.
New Definition 1.21 (Seam): a seam is the outer-boundary cycle
B_out^{(T)} of a non-root tread T, separating G into the seam
interior G_T and seam exterior G_C^{ext}. Notation Col(X | C) for
boundary-restricted 4-colourings is also defined here.
New Definition 1.22 (Partial tire tree): G_{T_r}^{circle} =
G_{T_r} with V(C_{T_r}) removed, i.e. the strict interior of the
triangulated disk inside the seam.
New Lemma 1.23 (Seam edges shared by <= one other depth-d seam):
an edge on the seam of a depth-d tread T is in the seam of at most
one other depth-d tread T'. Proof via inner-dual-of-outerplanar-
is-a-tree: C_T bounds a face of the parent's O^{(T_p)} (outerplanar),
so each edge of O^{(T_p)} lies in at most two of its bounded face
cycles, giving at most one sibling seam containing e.
New Conjecture 1.24 (Seam structure of minimum 4CT counterexamples,
sketch): a hypothetical minimum 4CT counterexample has bilateral
colourability, bilateral incompatibility, Birkhoff's seam-length
>= 6 bound, and an innermost obstruction at a leaf tread T^* whose
seam interior is one of a finite list of minimal seam configurations,
with the boundary palette restriction propagating outward along the
root-to-T^* obstruction chain.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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