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>
Concrete empirical example added to boundary_cut_tire.tex (page 2):
HM_0 cut #1 side 1, d=2:
- H_2 has 3 faces (lengths 4, 4, 12).
- H_1 has 3 faces (lengths 4, 4, 12).
- The length-12 H_2 face is low-side (contains pendants + H_1
edges in its interior).
- Adjacent H_1 edges come from ALL THREE H_1 faces:
H_1 face 0: edge (15,19)
H_1 face 1: edge (17,21)
H_1 face 2: edges (23,27), (28,33), (24,29), (28,34)
- No single H_1 face contains all of them → no unique parent.
This is a genuine empirical case, not a schematic. The figure
(uniqueness_break_example.pdf) shows the planar embedding from
Sage with:
- Orange = H_2 face 2 boundary (12 edges)
- Green / purple / blue = H_1 edges grouped by their H_1 face
- Gray = pendants (d=0) and depth-3+ edges
- Red dots = pendant vertices
Two new scripts:
- find_uniqueness_break.py: searches for empirical cases
- draw_uniqueness_break.py: renders the figure using Sage's
planar embedding
Note grows to 6 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Previous figure shaded all of B in red, making it look like X
and Y were the same red region. New version distinguishes:
- X (annulus between H_d and H_{d-1}) = light cyan.
- Y (exterior of H_{d-1}) = light yellow.
- A (inside H_d) = white.
- B (low-side face of H_d) = everything outside red dashed
boundary = cyan ∪ yellow.
Now visually clear:
- X is the cyan annulus (proper subset of B).
- Y is the yellow exterior (proper subset of B).
- B = X ∪ Y, neither X nor Y contains all of B.
The two arrows from "face B" label point at both regions to
emphasize B spans both faces of H_{d-1}.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Concrete picture demonstrating "low-side faces span multiple
parent faces of H_{d-1}":
- H_{d-1} drawn as the outer blue circle.
- H_d as a smaller nested orange circle inside.
- Face A of H_d (high-side, inside inner cycle): a small disk,
sits entirely inside face X of H_{d-1}. Unique parent. ✓
- Face B of H_d (low-side, outside inner cycle): RED REGION
spanning across the H_{d-1} cycle. It is one connected face
of ℝ² \ H_d, but it intersects BOTH face X (annulus between
cycles) and face Y (exterior of H_{d-1}). Neither X nor Y
contains all of B → no unique parent. ✗
This makes the uniqueness step's failure visible: the forest
proposition's containment argument works for high-side (= face A,
nested inside) but fails for low-side (= face B, spanning across).
The motivation section of the note now has 3 figures:
1. Low-side spans uniqueness failure (this commit, page 1)
2. T_∂ thick-H_1 hexagon (page 2)
3. T_∂ thin-H_1 tree (page 2)
Plus the nested-cut-tires figure on page 3.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW SECTION: "High-side cut tires alongside T_∂" with a stacked
figure showing concentric structure:
- Outer hexagon (blue) = T_∂'s cycle (depth 1 = ∂f_∂).
- Inner triangle (orange) = a depth-2 high-side cut tire
T_2^(f') interior to T_∂.
- Red dashed pendants outward from T_∂ = cut edges (depth 0,
OUT spokes of T_∂).
- Green dashed edges between outer and inner = depth-2 edges,
serving simultaneously as IN pendants of T_∂ and OUT spokes
of T_2^(f'). These are the SHARED EDGES the chain DP uses.
- Purple dashed pendants from inner triangle = depth-3 edges
(IN spokes of T_2^(f'), going to grandchildren).
The picture makes concrete what was abstract: each tire has
the same shape (cycle + 2 classes of spokes), tires nest
concentrically by depth, and the chain DP's edge-sharing is
literally the geometric "this same edge is on both tires."
Note grows from 4 to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two illustrations in boundary_cut_tire.tex:
1. Thick H_1 case: hexagonal H_1 with 3 OUT pendants (outward,
dashed red) and 3 IN pendants (inward, dashed green). Shows
T_∂ cycle as the hexagon boundary between low-side outer
region (red shade) and high-side inner region (green shade).
2. Thin H_1 case: H_1 = path of 3 edges (tree), single face is
f_∂, boundary walk traverses each edge twice (gray arrows
showing the two-sided traversal). 2 OUT pendants at V_{deg=2}
vertices, no IN pendants. This is the dodecahedron cut #0
side 0 structure where the high-side-only DP failed.
Note grows from 3 to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: boundary_cut_tire.tex (3 pages)
NEW SCRIPT: experiments/boundary_cut_tire.py
CONCEPT: T_∂^(i) per side i = the unique low-side face of H_1
(= face containing all pendants) treated as a virtual root tire.
- Cycle = boundary walk of f_∂ (depth-1 edges)
- OUT pendants = depth-0 cut edges in f_∂'s interior
- IN pendants = depth-2 edges at boundary vertices going into
adjacent high-side faces
T_∂ adjoins the high-side forest as a boundary node: not parent
or child geometrically, but shares edges with adjacent high-side
tires (depth-1 boundary edges, depth-2 in-pendants).
The extended chain DP includes T_∂ and uses edge-sharing
compatibility with adjacent high-side tires.
EMPIRICAL RESULTS (vs. ground truth from brute-force enumeration):
Dodecahedron:
- cut #0 side 0 (|S|=4, H_1 = tree): MATCH 9=9 ✓
[previously high-side DP gave 0, framework failed]
- cut #3 side 1 (|S|=4): MATCH 9=9 ✓
- cut #4 side 0 (|S|=4): MATCH 9=9 ✓
- HM_0 cut #0 side 0 (|S|=4): MATCH 9=9 ✓
Thicker sides: |R_dp| < |R_ground| (DP over-restricts).
This is a separate issue (probably heuristic parent-finding
or shared-edge logic when multiple high-side tires interact),
not the coverage gap.
Some cuts have side too large for brute-force enumeration in
T_∂ (n_edges > 18 limit), marked 'bdy too big'.
KEY WIN: the coverage gap is closed for the thin-side case where
H_d is a tree. The boundary cut tire converts these from
"framework gives R=0" to "framework gives R = ground truth."
NOT YET CLOSED:
- Thicker sides where DP under-counts vs ground truth
(different sets, similar cardinality sometimes)
- Branched per-tire half (T_∂'s cycle can traverse edges twice)
- Strong per-tire extendibility conjecture
But the framework now has principled coverage on ALL sides,
not just those with cycles in H_1.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW: chain_dp_joint.py — chain DP tracking full per-tire colorings,
edge-tuple-based parent/child sharing, and ground-truth comparison
against brute-force G' edge-coloring enumeration.
KEY EMPIRICAL FINDING (4th issue in chain_half_analysis):
When H_d is a tree (no internal cycles), the high-side cut tire
forest is EMPTY. The single H_d face is forced (by the level-set
lemma) to be entirely low-side or high-side; for a tree containing
the pendants, it's low-side. Hence high-side forest has 0 tires.
This happens at dodecahedron cut #0 side 0 (|S_0|=4):
- depths {0: 2, 1: 3}, |H|=6, |E(H)|=5
- H_1 is a tree, 1 face of length 6 (= low-side)
- No high-side cut tires
- DP gives R_dp=0, but ground truth R=36
DP correctly produces non-empty output on side 1 (where H_1 has
2 faces, one high-side), but the high-side framework's coverage
is incomplete for thin (small |S_i|) cuts.
This is a STRUCTURAL gap, not a code bug. The path forward
suggested in chain_half_analysis.tex: introduce a "boundary cut
tire" T_0 representing the low-side face + its pendants, so the
chain DP runs from leaves through T_0 to the cut.
Compounding with prior gaps:
(1) cut tires aren't always spoke-only (branched H_d faces)
(2) OUT-only projection loses S_3 orbit
(3) heuristic parent-finding (vertex overlap)
(4) tree H_d → empty high-side forest (this commit)
Net: the loose conjecture's chain half is genuinely open and
requires framework extension before the DP can be tested cleanly.
S_3 equivariance and high-side forest structure are the proven
pieces.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW NOTE: chain_half_analysis.tex (4 pages).
Formulates the chain half of the loose conjecture as a tree DP
over the cut tire forest, identifies what's proven vs. open:
PROVEN
- Tree structure (high-side forest): from
cut_tire_tree_structure.tex
- S_3-equivariance of the DP: trivial Lemma in this note
- Per-tire half for spoke-only cut tires (n ≥ 3): Prop 1.13
OPEN / GAPS DISCOVERED
1. Cut tires are NOT in general spoke-only. H_d can have degree-3
vertices (= branch points), making face boundaries non-simple
cycles. Dodecahedron 6-edge cut yields H_1 with one face of
length 20 over only 11 distinct vertices. Prop 1.13's count
2^n + 2(-1)^n applies only to spoke-only tires.
2. OUT-only projection loses S_3 orbit info. The per-tire half
guarantees a full S_3 orbit on the JOINT (in + out) projection,
but restricting to OUT spokes can collapse to |A|=3 (constant
tuples). Empirically observed ~20% of the time on test cases.
Correct DP must track joint projection (analog of
tire_fiber_step2.tex's joint-support tracking).
3. Non-emptiness preservation through the DP is the genuine open
piece (Conj. in this note + Strong per-tire extendibility).
EMPIRICAL TESTS
- chain_dp_test.py: simple cycle DP (assumes spoke-only).
- chain_dp_general.py: handles branched faces via brute-force
3-edge-coloring enumeration (cut off at 12 edges/tire for
tractability).
- chain_dp_debug.py: diagnostic for inspecting H_d face structure.
The general test reveals all three gaps above when run on
Dodecahedron + HM #0. Cross-cut R_0 ∩ R_1 should be non-empty
for both (they are 3-edge-colorable), but the heuristic
parent-finding plus OUT-only projection produce false negatives.
Status table at end of note summarizes what's needed to close.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the informal Stage 2 argument with a rigorous one,
achieved by refining the proposition to high-side faces only.
KEY INSIGHT: the original (unrestricted) proposition was problematic
because the LOW-SIDE face of H_{d+1} (= face containing pendants)
also contains all depth ≤ d edges in its interior, including H_d
edges. Hence low-side H_{d+1} faces span multiple H_d faces.
The fix: restrict to HIGH-SIDE faces only.
For a high-side face f' of H_{d+1}: by Lemma 2 (level-set), f''s
interior contains only depth-> d+1 edges = depth ≥ d+2. Since
depth-d edges are NOT in this range, no H_d edge sits inside f'.
Therefore f' is contained in a unique H_d face (by partition).
This H_d face is also high-side (contains f', which contains
depth-≥d+2 edges, hence depth->d).
Result: high-side cut tires form a forest, rigorously. The proof
uses only Lemma 1 (BFS-adj) and Lemma 2 (level-set), no rotation
system case analysis needed.
Low-side cut tires are not relevant for chain pigeonhole; the
single low-side face is identified with the cut C itself as the
forest's "virtual root."
Note grows to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the earlier sketch with a more detailed two-stage proof:
Stage 1: BFS level-set lemma.
Lemma (BFS-adj): adjacent edges in line graph differ in depth by
at most 1. Proof: BFS-distance property.
Lemma (level-set): every face of H_d contains only edges of depth
<d, or only edges of depth >d. Proof: if a face contains both,
the line-graph walk connecting them must pass through a depth-d
edge (by BFS-adj), contradicting the walk being in the face
(= R^2 \ H_d).
Stage 2: faces of H_{d+1} embed in faces of H_d.
Key claim: no H_d edge sits strictly inside any face of H_{d+1}.
Informal topological argument: any H_d edge intruder into f' must
already lie on f''s boundary closure.
Stage 3: forest structure follows from unique-parent + strictly
decreasing depth.
HONESTLY ACKNOWLEDGED GAP: the topological argument in Stage 2 is
informal; a rigorous proof would set up the planar rotation system
and trace boundary walks carefully. Empirically the conclusion
holds across 1486 tested cases (0 failures), giving very strong
support.
Note grows to 5 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Strictly tests the cut-tire forest property on cubic plane graphs
whose primal triangulation is internally 6-connected (= eligible
to be a minimum counterexample to the 4CT, per Birkhoff 1913).
Verified internal 6-connectivity of two primal triangulations
(exhaustive check over all 5-vertex subsets):
- Icosahedron (12v, 5-regular): YES, internally 6-connected.
Dual = Dodecahedron.
- Pentakis dodecahedron (32v, min deg 5, max deg 6):
YES, internally 6-connected. Dual = BuckyBall.
Tree structure sweep on the corresponding duals:
- Dodecahedron: 45 cuts, 45/45 produce trees on both sides.
- BuckyBall (60v cubic plane): 60 cuts, 60/60 produce trees.
- TruncatedTet (12v): 2 cuts, 2/2 produce trees.
105/105 cuts on minimum-counterexample-eligible duals produced
trees on both sides. 0 failures.
(Tutte graph: ran out of timeout enumerating its 6-edge cuts;
skipped from final tally.)
This is the most direct evidence for Proposition (cut tires form a
forest): the tree structure holds on the actual Birkhoff-eligible
graphs.
Files:
experiments/eligible_sweep.py
experiments/eligible_sweep_data.txt
notes/cut_tire_tree_structure.tex (updated)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds tree_structure_sweep.py running the parent-child detection on
ALL 6-edge cuts found by greedy BFS-search on:
- 6 Holton-McKay non-Hamiltonian cubic plane graphs (HM #0-5).
- Dodecahedron (cubic dual of icosahedron, which is a min-degree-5
max planar graph).
Total 743 distinct 6-edge cuts × 2 sides each = 1486 tests.
Total cut tires examined: 11,477.
Tree-structure failures (cycles in parent relation): 0.
Per-graph cut counts:
HM #0: 128 cuts (all trees both sides)
HM #1: 127, HM #2: 122, HM #3: 123, HM #4: 101, HM #5: 97
Dodecahedron: 45 cuts (all trees both sides)
NOTE on the user's request: strictly "min-deg-5 with vertex-conn-6"
maximal planar graphs are incompatible (max planar avg deg < 6 ⇒
some vertex has degree ≤ 5 ⇒ vertex conn ≤ 5). Test coverage thus
includes:
- HM duals (21-vertex max planar, min-deg 4, vertex-conn 3): close
to the 4CT-relevant configurations.
- Icosahedron (12-vertex 5-regular, vertex-conn 5): min-deg 5
case.
Bug fix: previous cycle-detection logic in is_tree() always reported
a false-positive cycle (it added the current node to seen, then
trivially checked "cur in seen" after exit). Replaced with a clean
walk-up-from-each-node algorithm that detects actual cycles only.
Adds:
experiments/tree_structure_sweep.py
experiments/tree_structure_sweep_data.txt
Updates notes/cut_tire_tree_structure.tex with broader sweep table
and totals. Note grows to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
User observation: the cut tires can at most have a tree structure.
This is correct: each face of H_{d+1} lies inside exactly one face
of H_d in the planar embedding, giving a parent-child relation that
is a forest (rooted at depth-1 cut tires).
PROPOSITION: parent(T_{d+1}^{(f')}) = T_d^{(f)} where f is the
unique face of H_d containing f' in its interior. Well-defined and
unique because H_d's faces partition the plane minus H_d's edges.
CONSEQUENCE FOR CHAIN HALF: chain pigeonhole reduces to a tree-DP
problem. Process tires bottom-up from leaves; at each node, combine
with children via the in-spoke ↔ face-boundary-edge bijection;
at the root, R_i is the projection. Tree DP is well-understood;
counterexamples (if any) must come from tree-DP failures, which is
much narrower than general-graph compatibility.
EMPIRICAL CHECK on G'_1 of HM#0:
Root (1, 0): |f|=12, no children (outer shell).
Root (1, 1): |f|=4, deep substructure all the way to depth 7
with single chain of children.
EMPIRICAL CHECK on G'_0:
Root (1, 0): |f|=9, one depth-2 child.
Root (1, 1): |f|=9, no children.
In both cases the structure is a tree (= 2-root forest).
CAVEATS:
- The empirical parent test used a vertex-sharing heuristic that
gives ambiguous candidates in some cases (8 ambiguous in G'_1).
A rigorous test would use point-in-region containment via the
planar embedding's face structure.
- The proposition itself is uncontested; the ambiguity is just an
artifact of the empirical detection.
NEXT STEPS:
1. Prove the proposition rigorously via point-in-region.
2. Implement tree DP on the cut tire forest.
3. Bound |R_i| as a function of tree size.
Note: cut_tire_tree_structure.tex (4 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adopts the k>=2 refinement of the loose chain pigeonhole conjecture
(per loose_conjecture_counterexamples.tex) and runs a broader sweep:
- All 6 Holton-McKay non-Hamiltonian 38-vertex cubic plane graphs.
- 3 candidate matching 6-edge cuts per graph (greedy search,
preferring matching cuts then balance).
- Both sides of each cut.
- All depths d >= 1.
- Brute-force enumerate proper edge 3-colorings (skipping cut
tires with > 14 edges due to runtime).
Results:
- 287 total cut tires examined.
- 154 with k >= 2 in/out spokes.
- 107 verifiable (≤ 14 edges).
- ALL 107 passed: |π(T)| >= 6 with at least one full S_3-orbit.
- 0 counterexamples found.
This is strong empirical support for the k>=2 form of the loose
conjecture's per-tire half.
The cut_depth_label note (now 7 pages) is updated with:
- k >= 2 restriction in the conjecture statement.
- Restriction rationale (k=1 trivially excluded).
- Status: empirical sweep + provable spoke-only case.
Files:
experiments/loose_conjecture_sweep.py
experiments/loose_conjecture_sweep_data.txt
notes/cut_depth_label.tex (updated)
Next step: the per-tire half is essentially provable for spoke-only
cut tires via Prop 1.13. The chain half remains the genuinely open
piece.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis