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>
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>
Moved from coloring_nested_tire_dual_graphs/ TO coloring_nested_tire_graphs/:
- Proposition (Source-side simple-cycle property) → now 1.7
- Lemma (Tire-component lemma) → now 1.8
- Theorem (Tire treads partition the bounded faces) → now 1.9
- Remark (boundaries-may-be-degenerate) → now 1.10
- Remark (no extra hypotheses needed) → now 1.11
These are foundational structural results about tire-graph
decompositions induced by a level source, not specifically about
the partial tire dual D(T) or coloring. Belongs in the
foundational paper.
Updates:
- Internal \cite[Definition~1.5]{bauerfeld-nested-tires} inside
the moved blocks → local \ref{def:tire-graph}.
- Foundational paper abstract rewritten to highlight the
tire-component lemma and tread partition as the main results.
- Dual paper abstract trimmed: no longer claims the tire-component
lemma as its own contribution.
- Dual paper intro citation list adds bullets for the moved
lemma (\cite[Lemma~1.8]) and theorem (\cite[Theorem~1.9]).
- No external references to the moved items inside the dual paper.
Page counts:
- Foundational: 3 → 7 pages.
- Dual: 9 → 7 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW: Theorem 1.5 (Tire treads partition the bounded faces).
For a maximal planar graph G with level source S on the outer face,
the family of tire treads { R_{C'} : d ≥ 0, C' a connected component
of G'_d } supplied by the tire-component lemma partitions the
bounded part of |Π_G|:
(i) every bounded face of G lies in exactly one tread R_{C'};
(ii) distinct treads have disjoint interiors.
Proof: each bounded face has a unique dual depth d, hence its dual
vertex lies in G'_d alone, and within G'_d in a unique component C'.
By the tire-component lemma, that C' carries the unique tread
containing the face.
This is the first step toward a chain pigeonhole argument that
colorings extend across the nested tire treads induced by a level
source.
Paper grows to 10 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Foundational paper: Definition 1.5 (Tire graph) now explicitly
names the closed planar region R bounded by B_out and B_in the
"tire tread of T". Remark 1.6 and the abstract updated to use
the new term.
Dual paper: places that referred to R as "the closed annular
region" or "the annular region" updated to use "tire tread" for
consistency:
- Definition 1.1 (Partial tire dual)
- Caption of figure on partial tire dual example
- Two places inside the proof of Proposition 1.2
"annular edges" (E_ann) and "annular faces" (F_ann) kept as-is
since they're established notation; the tread is the region they
triangulate.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The previous abstract/intro still treated "partial tire dual" as
foundational vocabulary defined elsewhere. After moving Definition
1.7 into this paper, the wording is fixed:
- Abstract: now lists tire graphs + dual depth as foundational
(from companion paper), and notes we DEFINE partial tire dual
here.
- Intro: removes "partial tire duals D(T)" from the list of
foundational vocabulary cited from the companion paper.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
REMOVED from coloring_nested_tire_graphs/:
- Definition 1.7 (Partial tire dual)
- Figure 3 (partial tire dual example)
- Figure 4 (partial tire dual bridge case)
- fig_partial_tire_dual.png file
- fig_partial_tire_dual_bridge.png file
- Abstract no longer mentions partial tire dual
Foundational paper now ends at Remark 1.6 (tire face/edge counts).
Down from 5 to 3 pages.
ADDED to coloring_nested_tire_dual_graphs/:
- Definition (Partial tire dual) — now numbered 1.1 in this paper
- Figure: partial tire dual example
- Figure: partial tire dual bridge case
- Both PNG figure files
Inserted before the structure proposition (former 1.1, now 1.2).
Intro citation list removes the bullet for partial tire dual since
it's now defined locally. The definition's internal ref to
Definition~\ref{def:tire-graph} becomes
\cite[Definition~1.5]{bauerfeld-nested-tires}.
The two figure captions updated to reference
prop:partial-tire-dual-structure locally (instead of citing the
companion paper as if it owned the definition).
Paper grows from 8 to 9 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW PAPER: papers/coloring_nested_tire_graphs/ ("Coloring Nested
Tire Graphs", 5 pages).
Contains foundational definitions 1.1 through 1.7 from the dual
paper, plus the four illustrative figures:
- 1.1 Level source
- 1.2 Levels
- 1.3 Dual (with label def:dual added — was missing in original)
- 1.4 Dual depth
- 1.5 Tire graph
- 1.6 Remark (tire counts)
- 1.7 Partial tire dual
Also: the dual-depth figure, the tire-example figure, and both
partial-tire-dual figures (vanilla + bridge case).
MODIFIED: papers/coloring_nested_tire_dual_graphs/paper.tex now a
follow-up:
- Abstract recasts the paper as building on the foundational paper.
- Intro no longer recapitulates definitions; lists them as
citations to the new paper.
- Removes definitions 1.1-1.7 and their figures (now in
foundational paper).
- Internal \ref{...} to removed labels converted to
\cite[Definition N.M]{bauerfeld-nested-tires}.
- Bibliography adds the new paper as a reference.
- Renumbering: theorems/propositions now start at 1.1 (formerly
1.8). Paper down from 14 to 8 pages.
Both papers compile cleanly with no broken references.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Renames the paper directory and updates:
- \title in papers/coloring_nested_tire_dual_graphs/paper.tex:
"Coloring Nested Tire Graphs" → "Coloring Nested Tire Dual Graphs"
- bibliography reference in papers/plane_depth/paper.tex
- rebuilt both PDFs
The new title reflects that the paper studies the cubic DUAL of
maximal planar graphs (nested tire structure on G^*), not the
primal triangulation.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Previous version had loose formulas and overstated what second-link
length forces. Replaced with cleaner version that:
- States the maximal-planar constraints explicitly
(E = 3V-6, F = 2V-4, sum of deg = 6V-12).
- Notes the FORCED 12 degree-5 vertices when all degrees ∈ {5,6}.
- Gives the correct second-link length formula:
L_2(v) = d + sum_{u in link(v)} (deg(u) - 5)
Earlier version had this wrong.
- Concretely: pentakis dodecahedron has L_2 = 10 around every
vertex, but its dual (Buckyball) STILL has 6-edge cyclic cuts
arising from non-second-link constructions.
So second-link length being large doesn't prevent small non-facial
cyclic cuts via other separators. The min cut size is not pinned
down by local link structure alone.
Bottom line unchanged: min non-facial cyclic cut for a min 4CT
counterexample could be 6, 7, 8, ... and Birkhoff alone doesn't
distinguish.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Added section "Could the minimum non-trivial cyclic cut be 8?"
Answer: yes in principle. Birkhoff gives "≥ 6"; nothing in the
condition pins the value to 6. A planar cubic G^* with min
non-facial cyclic edge connectivity = 8 would have:
- No non-facial 6-edge cut.
- No non-facial 7-edge cut.
- Some non-facial 8-edge cut.
By cut-parity lemma: 8-cuts have even sides; 7-cuts have odd
sides.
Heuristic when this happens: link vertices of degree ≥ 6
(rather than icosahedron-tight 5) push second-link length to
≥ 10, eliminating small non-facial separators.
The cut-tire framework adapts: chain DP and T_∂ construction
work for any cut size, with parameter changes. Per-tire half
needs re-examining for larger structures.
Bottom line: min non-trivial cyclic cut is one of {6, 7, 8, 9,
...}; Birkhoff doesn't pin it down. Framework's natural domain
is whatever value it happens to take, with 6 being the simplest.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
UPDATED: birkhoff_internally_6_connected.tex now adds the distinction
between "internally 6-connected" (= cyclic edge conn ≥ 6 in dual)
and the framework's needed condition (= cyclic edge conn EXACTLY 6,
so 6-edge cuts exist). Notes that this is a real a priori
restriction not provided by Birkhoff alone.
NEW NOTE: even_separating_cycle.tex (3 pages)
Addresses: "must a min 4CT counterexample have a separating n-cycle
with n even and n ≥ 6?"
Honest answer: I don't know of a proof either way.
Key contributions:
- Lemma (cut-parity in cubic graphs): |C| ≡ |S| ≡ |T| (mod 2).
So even-length cycles in primal G ↔ cuts with even-sized sides
in dual G^*.
- |V(G^*)| = 2|V(G)| - 4 is always even, so both sides have
matching parity.
- Birkhoff doesn't rule out odd-length separating cycles ≥ 7.
- Second-link heuristic: in internally 6-conn triangulations,
the "second link" of any vertex is typically a 6-cycle, giving
abundant separating 6-cycles in practice. But this is
heuristic, not proven for all such triangulations.
Conjecture (stated, not proven): every internally 6-conn planar
triangulation with ≥ 12 vertices has a separating even n-cycle
with n ≥ 6.
Equivalent: every planar cubic graph with cyclic edge connectivity
≥ 6 and ≥ 20 vertices has a cyclic edge cut of size exactly 6.
This is a structural question; I don't know a planar cubic
counterexample.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: birkhoff_internally_6_connected.tex (3 pages)
NEW SCRIPT: experiments/draw_internally_6_connected.py
NEW FIGURE: icosahedron_internally_6_connected.pdf
States and illustrates the Birkhoff (1913) condition that any
minimum 4CT counterexample must be internally 6-connected:
- No separating 3-cycle.
- No separating 4-cycle.
- No separating 5-cycle isolating ≥ 2 vertices on either side.
- Only separating 5-cycles isolating exactly 1 vertex.
The icosahedron is the canonical example: 12 vertices all of
degree 5; the 5 neighbors of every vertex form a 5-cycle whose
removal isolates that vertex. Sage verification confirms this:
Vertex 0 has 5 neighbors: [1, 5, 7, 8, 11]
Induced subgraph on neighbors: 5 edges, is_cycle=True
After removing the 5 neighbors: 2 components, sizes=[1, 6]
Note also lists the graphs used in our framework testing:
- Icosahedron (12 v, dual = dodecahedron)
- Pentakis dodecahedron (32 v, dual = Buckyball)
- Holton-McKay graphs (21 v primal, 38 v dual)
All are internally 6-connected, hence in the framework's intended
domain.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis