Full uniform-family CSP at n=14 --no-tri (4403 tiles, 193 branching) is FEASIBLE:
one family threads every tile incl. branching nodes (outer rim + both inner faces
at once). Independent candidate test threads 193/193 branching tiles. Witness is
fully regular: sigma_m = 0^m if m even (monochromatic), 0^(m-2)12 if m odd. So on
the 4CT-relevant class the chained pigeonhole is constructively resolved throughout
the tested range (n=9,12,14, incl. branching).
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add kempe_branching_min_probe.py (structural: >=2 inner faces with singletons).
Unrestricted branching first appears at n=11; no-separating-triangle branching
(>=2 inner faces each >=4 singletons, p>=4) first appears at n=14 (193 tiles).
Smallest example: word=UUUUDDDDDDDDDD bite=(8,13), p=4, faces root{4,5,6,7} and
bite{9,10,11,12}. n=14 is the smallest place to test the uniform family / R_T
composition on a genuine branching no-separating-triangle tile.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add --no-tri filter (exclude tiles with a length-3 boundary = separating/non-facial
triangle in G: outer rim of 3 up teeth, or an inner face of exactly 3 singleton
downs) to the trend and uniform-family probes.
The n=12 breaker UUUDUDUDUDUD bite=(3,11) has a size-3 inner face (encloses d5,d7,d9)
and is excluded. With the restriction the size-7 universal at n=12 is restored
(|D[7]| 0->2), every |D[m]|>=1 across n=6..13, and the uniform-family CSP becomes
FEASIBLE at n=12 with the simplest family (monochromatic on even sizes, min-cut on
odd). So the only universal failure was an artifact of admitting non-4-connected
configs; on the 4CT-relevant class gluing is constructively trivial in range.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add plot_breaker_tile.py and figure for word=UUUDUDUDUDUD bite=(3,11): structure
(7 up teeth = size-7 outer rim, bite (3,11), singleton downs d5,d7,d9) plus a
Kempe-balanced colouring. Reconfirms the outer rim realises 9/10 admissible size-7
necklaces, never 0001112 -- the lone tile that empties the size-7 universal at n=12.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add kempe_universal_trend_probe.py (|D[m]| per size across n). Across n=6..13 and
all sizes, the ONLY empty per-size universal is (n=12, m=7): at n=13 size 7 is back
to |D|=2 with more boundaries (579), so the vanishing is sporadic, not monotone.
The lone n=12 breaker is the outer rim of word=UUUDUDUDUDUD bite=(3,11) (most-
alternating 7-up word, antipodal bite), realising 9/10 size-7 necklaces and missing
only 0001112. Correct the earlier "doomed at scale" reading in the findings note:
the uniform shortcut almost always works (near-total coverage) but is fragile to a
single exceptional tile; pairwise gluability still always holds.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Write up the R_T coupling, the uniform-family result (feasible n=9, infeasible
n=12 via empty size-7 universal, 0001112 blocked 210/211), and the open threads.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Pursue the paper's medial pigeonhole programme (R_T restriction relation,
chain-pigeonhole conjecture) at the data level.
Findings: R_T (outer<->inner boundary necklace, one Kempe-balanced colouring)
is genuinely coupled, not a product of its projections. A uniform per-size
boundary-state family threading every tile EXISTS at n=9 (unique per size, the
balanced-block necklaces 0011/000011/012/00012 -- not monochromatic), but FAILS
at n=12: size-7 seams admit no universal state (|D[7]|=0; near-universal 0001112
realised on 210/211 boundaries, blocked by one tile). So the uniform "same state
everywhere" shortcut breaks once large odd seams appear and universals vanish as
the tile population grows; the per-interface pigeonhole choice is genuinely
needed. Pairwise gluability still holds, so this locates the conjecture's
difficulty rather than obstructing gluing.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Each tile realises only a subset of the parity-admissible alphabet on its rim,
and tiles genuinely omit interfaces (n=12 m=8: max 273/274, min 43). But any
two tiles always glue: interface subsets always overlap (n=9 m=3-6, n=12 m=3-8)
-- usually via a global universal seam present on every inner+outer rim, and
where none exists (n=12 m=7) the worst pair still shares 14 seams. The universal
seams are the low-complexity ones (<=2 colours, single contiguous block). No
local gluing obstruction; any obstruction must be global across a nested stack.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
For each interface size m, compare the realized census vocabulary (outer
up-tooth apexes and inner singleton-down apexes) against the full
parity-admissible set. At n=12, m=3..8 every parity-admissible sequence is
realized on both faces (counts 1,4,10,31,91,274; none missing), and up==down
throughout -- the n=9 result is n-independent and scales to m=8. Validated
against the known n=9 answer before running n=12.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
The anchored single-representative reading interacted with dihedral graph
dedup to record an arbitrary orientation of each necklace, producing a
spurious up-vs-down split at n=9,m=6 (001212 only up, 010122 only down --
the same necklace). Add dihedral_reading_sequences(), which unions the
canonical reading over all 2n dihedral anchors and exactly reproduces the
brute un-deduped census; make it the default for both experiments, with
--anchored to recover the old behaviour. Document the artifact and fix in
kempe_sequence_orientation_note.md.
Regenerate up + down for n=9, m=3..6. Up and down now agree on sequences
and groupings at every m (m=6: identical 31 sequences, 6 groups; the
001212/010122 pair appears on both sides). Groupings coarsen vs anchored
(m=4: 3 groups; m=5: 2 groups) since the orientation-honest vocabulary
merges previously split sequence-sets.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Mirror of the up-tooth experiment with the distinguished valid face moved
from the outer face to an inner non-tooth face (root or bite inner-gap).
For each (M(T), inner face) config holding m singleton down-tooth apexes,
record the apex colour sequence (cyclic order, mod colour permutation) over
Kempe-balanced colourings and group configs by their sequence-set. Runs for
m=3,4,5,6 with per-sequence notes, figures, and a config atlas.
Finding: inner faces realise the same parity-admissible sequence vocabulary
and the same distinct-sequence counts (1/4/10/28) as the outer face, i.e.
the Kempe-parity law acts uniformly on every valid face. At m=6 the configs
are the U<->D embedding mirror of the up-m=6 graphs (matching 7 configs,
28 sequences, 127 colourings).
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Enumerate Kempe-balanced 3-colourings of every M(T) with |A(T)|=9 and a
fixed number m of up teeth, record the up-tooth apex colour sequence
(cyclic order, mod colour permutation only), and group the M(T) by their
set of unique sequences. Runs for m=3,4,5,6 with per-sequence notes and
figures plus a summary atlas.
Finding: realised sequences obey outer-face Kempe parity (all three
colour-counts share m's parity). Distinct sequences grow 1/4/10/28 while
M(T) count falls 23/29/18/7 across m=3..6.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add draw_tire_realization.py: for each full medial tire graph from the seed-1
analysis, draw every proper 3-colouring (mod colour permutation) in a grid,
each panel coloured by its three colour classes and banner-labelled Realized /
Unrealized / Invalid, and write one standalone note per graph (plus a README
index). Refactor tire_realization_analysis to expose iter_pieces() yielding
per-piece coloured colourings.
Output: tire_realization_seed1/ with 17 piece notes + figures.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
For a random 12-vertex maximal planar graph (sphere convex hull), enumerate
all proper 3-colourings of M(G), take the BFS-level (tire-tree) decomposition
from every source vertex, and build each full medial tire graph M(T) in the
ambient tread-face model (cycle + teeth + bites). Recognise each M(T) as a
FullMedialTireGraph and label every proper 3-colouring Realized (Kempe-balanced
and a restriction of a global colouring), Unrealized (balanced but not a
restriction), or Invalid (not balanced).
Findings on seed 1 (17 pieces, M(G) with 90 colourings): zero realized-but-
invalid colourings (confirms Remark 5.8 on a real triangulation), and 12 of 17
pieces carry Unrealized colourings -- Kempe-balance is necessary but not
sufficient for realization; it is sufficient only on cap-like all-up/shallow
treads.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Bites arise when the inner outerplanar graph O has a bridge: the bridge
edge is traversed twice by the outer-face walk, so its medial vertex is
adjacent to four annular vertices.
- check_remark58_bite.py: a minimal bite tread (outer 4-cycle + interior
bridge u-w) restricts to Kempe-balanced on all colourings (outer face).
- check_remark58_bite_rich.py: O = triangle abc + pendant bridge a-d gives
one bite plus three singleton down teeth in the bite's inner-gap face;
every restriction is Kempe-balanced (the three gap singletons are a
rainbow in every global colouring).
Update Remark 5.8's verification note: the bite case, including singletons
in the bite-gap face, is now confirmed.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Computational checks of the necessity of Kempe-balance (Remark 5.8):
- check_medial_face_parity.py shows the naive "even P-coloured vertices
per medial face" claim is false (odd vertex-faces on the octahedron and
stacked triangulations), so the original face-parity justification was
wrong.
- check_remark58_bitefree.py builds genuine bite-free tire pieces (capped
triangulated annuli) and confirms every proper 3-colouring of M(G)
restricts to a Kempe-balanced colouring (|A(T)|=6,8,10,12, all
colourings, zero failures).
Rewrite Remark 5.8 to cite the correct mechanism: the up/down apexes lie
on level cycles, and a P-Kempe cycle meets each level cycle in an even
number of P-coloured incidences (Lemma 5.6). Note the bite case is not
yet checked end to end.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define Kempe-balanced colourings of a full medial tire graph (Def 5.7):
for each valid face (outer face or interior non-tooth face of B(T)) and
each colour pair {a,b}, the number of tooth apexes incident to the face
coloured a or b must be even. Add Remark 5.8 (necessity: a colouring of
M(T) extends to M(G) only if it is Kempe-balanced) and rename Lemma 5.5
to "Kempe chains are cycles".
Add kempe_valid_colorings.py: enumerate all proper 3-colourings of a full
medial tire graph, label each Kempe-balanced/valid or invalid, and plot
them with the offending face's Kempe chains and odd apex set highlighted
on invalid panels.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Name A(T) the "annular cycle" (Thm 3.3, Def 3.4); clarify the bite-face
condition in Remark 3.8 to count down-tooth apexes interior to each face;
add the non-incidence stipulation for bite edges to Def 3.7.
Add an exhaustive generator over |A(T)| enforcing the 3.1-3.9 properties
(tooth word, non-crossing non-incident bites, >=3 up teeth, bite-face
condition), plus a plotting script and the n=9 atlas (81 dihedral classes).
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Adds two TikZ figures (boundary-state worst cases and annular cycle
counterexample), a new subsection on Kempe-cycle conservation across
medial tires, and the experiment scripts/findings for the medial tire
restriction search and annular cycle condition check.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
Introduction now positions the tire-tree decomposition against
Birkhoff, Tutte, Heesch, Robertson-Sanders-Seymour-Thomas, and
Dvorak-Lidicky's coloring count cones (closest modern parallel).
Floor-containment conjecture refuted at n=4 and n=6: explicit
counterexample colorings (1,2,1,2), (1,3,2,1,3,2), (2,3,2,3,2,3)
absent from non-floor supports. Skip-m=3 sweep through m=8 partial
still consistent with floor-stability-in-m.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Quotient (m, k, path, chords, cycle) by the order-2(m+k) dihedral
action on the rung sequence; exhaustive sweep over outer 3..7, inner
7..9 yields 19 distinct supports with a unique floor at 252/732
realised by m=3, k=7, single-chord 6-face.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The 14-vertex 3-connected triangulation at plantri index 263993 has
no vertex source admitting a witnessing 4-colouring; the level-cycle
conjecture still holds on it via v0=10. Reorder the section so the
level-cycle conjecture follows the failed inner-boundary refinement.
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
Introduce Conjecture 1.31 (tire inner-boundary three-colour) as a
decomposition-native weakening of the level-cycle conjecture 1.29:
every maximal planar graph admits a vertex source and proper 4-colouring
under which each tire inner boundary omits a colour. Remark 1.32 shows
inner boundaries are single-level cycles, so the vertex-source form of
1.29 implies it on 2-connected boundaries.
Extend check_level_cycle_three_color.py with --restriction inner-boundary
(reconstructs the tire-tree decomposition from the embedding; inner
boundary = level-(d+1) vertices of each depth-d dual component) and a
--min-connectivity flag for the 5-connected slice.
Verified: full census 4<=n<=13 (57716 triangulations) and 5-connected
slice 14<=n<=24 (9732 graphs) all admit witnesses; no counterexample.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
In the tire-component lemma the induced subgraph that becomes the tire
graph was named C, clashing with C used everywhere else for cycles
(seam cycles C_T, cycle graphs C_n, the seam cycle C in Def 1.16).
Rename it to T_{C'} throughout the lemma statement, its proof, and the
degenerate-boundary remark, so C/C'/C_T are uniformly reserved for
cycles and components.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define previously-implicit objects and unify conventions:
- define level sets L_d (and L_{<d}, L_{>=d}) in the Levels definition
- factor G'_d, F_{C'}, V_{C'}, R_{C'} into a standalone definition
before Prop 1.6, removing the forward reference
- name the annular faces F_ann and state the tire-graph tuple form
T = (B_out, O, E_ann) in the tire-graph definition
- ground the full tire dual D(T) where Gamma is introduced
- normalize tree superscripts (0)/(p)/(c) to the tire-symbol form
(T_0)/(T_p)/(T_c)
- resolve the boundary-count clash: use nu = |V(B_in)| (inner) and
mu = |V(B_out)| (outer) throughout, freeing n for |V(G)|
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Replace the stale two-paper list (kempe_style_search_for_smaller_contradiction
and plane_depth_labelling -- neither in the repo) with a table of the
12 papers actually present under papers/, with titles pulled from each
paper.tex. Also clarify that init_paper creates papers/<name>/ rather
than a top-level directory.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
Delete Definition 1.20 (iso of trees of tire treads), Conjecture 1.21
(universal nesting), Conjecture 1.22 (seam realizability), the
seam-construction figure inclusion, Remark 1.23 (nesting reduces to
seam), and Remark 1.24 (motivation / open questions). The paper now
ends after Remark 1.19 (tree-coloring-factorisation).
The fig_seam_construction.png file and its generator script remain in
the repo as assets; nothing in the paper currently references them.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Rewrite Conjecture 1.20 (universal nesting) with the iso notion fixed
to combinatorial with O preserved: rooted tree iso + plane-outerplanar
iso of O on each tread + child/face correspondence, with B_out
explicitly not required to match (essential for sub-tree embedding).
Factor the technical core out as Conjecture 1.22 (seam realizability):
for every k >= 3, exhibit a triangulated planar disk H_k with
boundary a k-cycle whose BFS-from-boundary tree of treads is iso to a
given T_1. Add Remark 1.23 stating that universal nesting reduces to
seam realizability by excise-and-glue using the existing structural
theorems.
Reworked Remark 1.24 (motivation) keeps the compositional-colourability
and universality bullets, and replaces the old open-questions paragraph
with three concrete subproblems: a candidate apex-removal construction
for the seam, 6-connectivity preservation as the relevant 4CT
subproblem, and a justification of why the weaker iso notion is
necessary.
Add fig_seam_construction.png (and the matplotlib script that generates
it) illustrating the seam construction on a 10-vertex G_1 with
T_1 a chain of length 3; the script asserts BFS-from-boundary in H_5
reproduces ell_{G_1} on V(G_1) \ {S_1}, giving a verified small
instance of the conjecture.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW Conjecture 1.19 (universal nesting of tire-tread trees,
sketch):
For any two rooted trees of tire treads T_1 = T(G_1, S_1) and
T_2 = T(G_2, S_2), T_1 NESTS into T_2:
Choose any tire T in T_2 and any non-trivial bounded face f of
its inner outerplanar graph O^(T). Then there exists a maximal
planar graph G̃ with level source S̃ such that:
(N1) T(G̃, S̃) contains T_2 as a sub-tree.
(N2) The sub-tree rooted at the new child of T at face f is
isomorphic to T_1.
Informally: any tree of tire treads can be inserted into any
non-trivial face slot of any other tree of tire treads. The
class of trees of tire treads is closed under composition by
face-slot insertion.
Followed by Remark 1.20 motivating the conjecture:
- Compositional colourability: if 4-colourability of G̃ follows
from 4-colourability of G_1, G_2 via parent-child consistency
(Remark 1.18 / former tree-coloring-factorisation), then 4CT
propagates through nesting. A min 4CT counterexample would have
to be irreducible under such nesting.
- Universality: trees of tire treads become a "term algebra" for
decomposing plane triangulations; coloring arguments can be
inductive on this algebra.
Open subquestions in remark:
- Precise notion of "isomorphic as rooted trees of tire treads"
(combinatorial vs geometric vs up to embedding).
- Constructive description of G̃ from G_1, G_2, f.
- Compatibility with Birkhoff's internally 6-connected condition.
Page count: 12 → ~13.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Removed Theorem 1.16 (the count formula for spoke-only and
single-chord cases). Folded the cycle formula 2^n + 2(-1)^n into
the surviving remark so the only retained content is the structural
observation:
- Tait reduces 4-coloring count to 3-edge-coloring count of Γ.
- For Γ ≅ C_n (spoke-only): cycle chromatic polynomial gives
2^n + 2(-1)^n.
- For Γ with chords, the count depends on chord structure
(nested vs. sequential etc.), not just (n, k).
- Always computable in linear time via tree decomposition
(outerplanar has treewidth ≤ 2).
Page count: 12 → 11.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW Theorem 1.15 (Tait correspondence for tires):
#{4-colorings of T} / |S_4| = #{3-edge-colorings of Γ} / |S_3|
That is, the number of 4-vertex-colorings of the tire T up to
color permutation equals the number of 3-edge-colorings of the
inner dual Γ up to color permutation.
Proof: standard Tait. Encode 4 colors as Z_2 × Z_2; define
χ*(e*) = c(u) + c(v) for each interior annular edge. The
triangulation constraint guarantees χ* is a proper 3-edge-coloring
of Γ; the lift c → χ* is 4-to-1 (global Z_2 × Z_2 translation).
Quotienting by |S_4| = 24 and |S_3| = 6 gives the stated equality.
NEW Theorem 1.16 (count formula):
(i) For spoke-only tires (Γ ≅ C_n):
#{proper 3-edge-colorings of Γ} = 2^n + 2(-1)^n.
(ii) For single-chord tires (Γ ≅ Θ(1, b, c), b + c = n):
#{proper 3-edge-colorings of Γ} = 6(α_b α_c + β_b β_c),
where α_L = (2^{L-1} + 2(-1)^{L-1})/3,
β_L = (2^{L-1} - (-1)^{L-1})/3.
Verification: Θ(1, 2, 2) = K_4 \ e gives 6.
Proofs:
(i) Standard chromatic polynomial of cycle at k = 3.
(ii) Transfer matrix on the two non-chord paths with chord
color fixed and endpoint configurations enumerated.
Remark 1.17: For more chords, the count depends on the chord
arrangement, not just (n, k). Two outerplanar graphs with the
same vertex and chord counts can have different 3-edge-coloring
counts. But linear-time computation via tree decomposition
(treewidth ≤ 2 for outerplanar) is always available.
Added Tait's 1880 paper as bibitem.
Page count: 11 → 12. Theorem 1.18 (tree structure) renumbered from
1.15 to 1.18 to make room.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW Theorem 1.15: The tire treads from a single-vertex level
source S = {v_0} form a rooted tree T(G, S) under face containment.
Statement:
- Root: the depth-0 tire tread T_0 with degenerate outer
boundary {v_0} (the apex tire, B_out = {v_0}).
- Parent: for any tire tread T_c at depth d ≥ 1, the unique
parent T_p at depth d-1 is the tire whose inner outerplanar
graph O^(p) has B_out^(c) as one of its bounded faces.
Equivalently, R_c lies inside this bounded face of O^(p).
- Children: bijection with bounded faces of O^(p) whose
interior contains depth-≥(d+2) vertices.
Proof structure:
1. Root well-defined: G'_0 is connected (fan around v_0), so
unique component → unique T_0.
2. Existence of parent: faces immediately outside B_out^(c) on
the S-side have depth d-1, lie in some component of G'_{d-1}.
3. Uniqueness: by Proposition 1.7 (source-side simple-cycle
property), B_out^(c) is a simple cycle, and the depth-(d-1)
faces around it form a single contiguous arc in the dual,
hence one component → unique parent.
4. Children description: bounded faces of O^(p) are in bijection
with deeper component-tires.
5. Tree property: parent map strictly decreases depth, hence
no cycles, hence rooted tree.
Plus two clarifying remarks:
- Remark 1.16: multiple children iff O^(p) has multiple bounded
faces with non-trivial interiors. Spoke-only case → exactly
one child.
- Remark 1.17: combined with Theorem 1.9 (partition) and
Theorem 1.12 (outerplanar inner dual), any coloring problem
on G factors through:
• local outerplanar coloring on each tread,
• parent-child consistency along shared B_out^(c) cycles.
This is the structural setup for the chain-pigeonhole program.
Page count: 10 → 11.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the spoke-only Figure 4 with a true barbell example:
Setup:
- B_out: hexagon u_0..u_5 (red).
- O = barbell: triangle {a_1, a_2, a_3} + triangle {b_1, b_2, b_3}
+ bridge a_3-b_1 (light red).
- 14 spokes triangulate the annulus into 14 annular triangles:
6 outer-cap + 6 inner-cap + 2 bridge-cap.
Dual placement is precise:
- All 14 blue dots at exact triangle centroids (via TikZ
barycentric cs).
- 13 edges of the Hamilton cycle wrap around the annulus
crossing each spoke.
- The bridge dual edge connects the two bridge-cap triangles
directly (dashed blue chord across the cycle).
Resulting Γ ≅ Θ(1, 7, 7): Hamilton cycle of length 14 with a
single length-1 chord. Outerplanar (the length-1 chord has no
internal degree-2 vertex, so no K_{2,3} minor).
This now properly demonstrates the chord arising from a real
bridge, exactly as the theorem and Remark 1.14 describe.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Previous Figure 4 had two bugs:
(1) Dual vertices were placed in arbitrary positions, not at
annular triangle centroids.
(2) The "bridge" chord didn't actually correspond to a bridge,
since B_in was drawn as a single hexagonal cycle (which has
no bridges). For a real bridge, O needs to be a barbell.
Redrawn as a clean spoke-only example:
- B_out: hexagon (6 outer vertices u_0..u_5, red).
- B_in: triangle (3 inner vertices w_0, w_1, w_2, light red).
- V(O) = V(B_in), no chord of O, no bridge.
- Triangulation: 9 spokes between outer and inner.
- 9 annular triangles: 6 "outer-cap" + 3 "inner-cap".
- Dual vertices placed using TikZ barycentric coordinates at
each triangle's exact centroid.
- Dual graph Γ ≅ C_9 (just a cycle, no chords for spoke-only).
The chord/bridge case isn't drawn directly in the figure but is
referenced via Remark 1.14, which already discusses the bridge
case (Θ(1,b,c) = Hamilton cycle + length-1 chord) textually.
This keeps the figure correct and unambiguous; readers wanting
the chord case can refer to the remark or the dual paper.
Page count: 9 → 10.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two TikZ figures added to the outerplanarity theorem:
Figure (Case 1, disk tread): apex v_0 at center, hexagonal
non-degenerate boundary (red), 6 spokes (grey) forming a fan of
6 triangles. Dual Γ (blue) is the cycle C_6 connecting the 6
triangle centroids. Outerplanar trivially.
Figure (Case 2, annulus tread): two concentric hexagons for
B_out and B_in, spokes + one extra "bridge-style" interior
annular edge. Dual Γ is a Hamilton cycle of length 12 around the
annulus, plus one chord (dashed). All vertices on outer face →
outerplanar.
Also corrected the Case 1 proof: the disk has a single interior
vertex (the apex), so the triangulation is a FAN around the apex
(not a polygon-triangulation with no interior vertices), and Γ
is a cycle of length k (not a tree). This is still outerplanar.
Added tikz + backgrounds library to preamble.
Page count: 8 → 9.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW Theorem 1.12: For any tire graph T, the inner dual Γ of its
tire tread (= subgraph of D(T) induced on interior dual vertices)
is outerplanar.
The theorem also gives a constructive characterization: Γ admits a
planar embedding as a (possibly non-simple) Hamilton walk through
every d_f, plus zero or more non-crossing chords.
Proof structure (constructive):
Case 1 (R is a disk, one boundary degenerate): the polygon
triangulation has no interior vertex, so its dual is a tree
(p-2 vertices, p-3 diagonals). Trees are outerplanar.
Case 2 (R is an annulus, both boundaries non-degenerate):
Step 1 - Cyclic ordering: cut R along any spoke e* to convert
the annulus into a closed disk. The disk boundary traverses
B_out + e* + B_in (reverse) + e*, yielding a cyclic sequence
S of annular faces with multiplicities (one per boundary edge,
+ detours for boundary-free faces).
Step 2 - Hamilton walk: consecutive entries of S share an
interior annular edge or coincide; the resulting closed walk
in Γ visits every d_f (using detours for the rare interior
annular triangles with zero boundary edges).
Step 3 - Non-crossing chords: remaining interior annular edges
become chords. Since the underlying E_ann edges in T are
non-crossing in the planar embedding, the chords are
non-crossing in Γ.
Step 4 - Outerplanar layout: place the |F_ann| vertices on a
circle in S-order, draw walk edges as the circle, chords inside.
All vertices on outer face → outerplanar.
Two remarks following:
Remark 1.13: spoke-only case is the classical Hamilton cycle
Γ ≅ C_{n+m} with zero chords.
Remark 1.14: bridge case (O with a bridge whose 2 incident faces
are annular) gives the theta graph Θ(1, b, c) — Hamilton cycle of
length n + m_∂ plus a single length-1 chord. The length-1 chord
contributes no degree-2 branch vertex to a K_{2,3} subdivision,
explaining why this is outerplanar despite being a theta graph.
Foundational paper grows from 7 to 8 pages.
This theorem unlocks the chain pigeonhole argument over tire
treads: each tread's coloring problem is on an outerplanar dual
graph, where the structure is locally tractable.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis