Add related-work section, refute floor-containment conjecture
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>
This commit is contained in:
@@ -327,6 +327,38 @@ def enumerate_level_cycle_supports(
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return supports, stats
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def check_floor_containment(
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supports: dict[frozenset[tuple[int, ...]], list[dict]],
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) -> None:
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"""Test whether the smallest-size support is a subset of every other support.
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The supports are fully normalized (orbits under S_4 x D_n), so set
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inclusion here is the right test for the "floor support is contained in
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every admissible support" conjecture.
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"""
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if len(supports) < 2:
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print(" containment: <2 supports, trivially contained")
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return
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keys = list(supports.keys())
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min_size = min(len(k) for k in keys)
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floors = [k for k in keys if len(k) == min_size]
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print(f" containment check: floor support (size {min_size}) vs {len(keys) - 1} others")
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for floor in floors:
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bad: list[tuple[int, tuple[int, ...]]] = []
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for other in keys:
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if other is floor:
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continue
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if not floor.issubset(other):
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missing = next(iter(floor - other))
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bad.append((len(other), missing))
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if not bad:
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print(f" floor IS a subset of all {len(keys) - 1} other supports")
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else:
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print(f" floor is NOT a subset of {len(bad)} of {len(keys) - 1} other supports")
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for size, missing in bad[:5]:
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print(f" missing example: floor coloring {missing} absent from support of size {size}")
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def summarize(
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n: int,
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supports: dict[frozenset[tuple[int, ...]], list[dict]],
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@@ -357,6 +389,7 @@ def summarize(
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print(f" most restrictive support : {min_size}/{all_colorings} ({len(min_supports)} support types)")
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print(f" least restrictive support : {max_size}/{all_colorings}")
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print(f" overlap among max-restrict : min {min(pair_overlaps)}, max {max(pair_overlaps)}")
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check_floor_containment(supports)
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print(" examples:")
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printed = 0
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@@ -239,3 +239,73 @@ opens the door to closed-form support computation for the worst case.
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The obstruction to a quick proof is exotic `m ≥ 4` configurations not
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reachable by subdivision from an `m = 3` floor witness — none has
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appeared empirically, but their non-existence is not yet ruled out.
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#### Skip-`m=3` sweep (partial)
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To probe the conjecture more directly we ran a "skip-`m=3`" sweep
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explicitly excluding `m = 3` from the search:
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```bash
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python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \
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--n-min 6 --n-max 6 --outer-min 4 --outer-max 10 --inner-min 7 --inner-max 9 \
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--progress
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```
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The sweep ran for `11h 49m` and was terminated partway through `m = 8,
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k = 9` at chord-set #6 of the 2-chord block (raw=6.45M canonical=223k).
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Throughout the run --- across `m ∈ [4, 7]` exhaustively and `m = 8`
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partially --- the distinct-support count stayed locked at **`13`**
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(vs `19` for the full `m ∈ [3, 7]` run). The six missing support
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types are precisely the ones with `m = 3`-only witnesses, including
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the unique `252/732` floor.
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Conclusion of the partial run: no support of size `≤ 252` appeared
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under `m ≥ 4` in any configuration searched, and the support count
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saturated early, suggesting the support landscape for `m ≥ 4` is
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strictly looser than for `m = 3` even as `m` grows. The conjecture is
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not falsified by any data we have; pushing `m` to `9` or `10` is
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left as future work (path-count growth makes it expensive: `m = 10,
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k = 9` alone has `C(19, 10) ≈ 92{,}000` paths per chord set).
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### Floor-containment conjecture — refuted
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Floor-stability bounds the *size* of the worst-case support but not
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its content. For the tire-tree compatibility argument, what would
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have closed the loop is a containment statement:
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> **(Conjecture, refuted)** For every `n`, the unique floor support
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> `F_n` is a *subset* of every admissible level-cycle support. Every
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> level-cycle coloring that the `m = 3` floor witness allows is also
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> allowed by every other tire.
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The `check_floor_containment` helper in `level_cycle_support.py`
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tests this directly on the computed supports. The conjecture **fails
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at `n = 4` and `n = 6`**:
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```text
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n=4: floor (size 60) is NOT a subset of 1 of 2 other supports
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missing: floor coloring (1, 2, 1, 2) absent from support of size 72
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n=6: floor (size 252) is NOT a subset of 11 of 16 other supports
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missing: floor coloring (1, 3, 2, 1, 3, 2) absent from support of size 708
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missing: floor coloring (2, 3, 2, 1, 3, 1) absent from support of size 492
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missing: floor coloring (2, 3, 2, 3, 2, 3) absent from support of size 720
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```
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(`n = 3` is trivial — only one support type. `n = 5` happens to
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satisfy containment in the searched window but only against a single
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other support, so the data is too thin to be evidence either way.)
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The missing colorings cluster on **bipartite alternations**
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(`(1,2,1,2)`, `(2,3,2,3,2,3)`) and **3-periodic patterns**
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(`(1,3,2,1,3,2)`). These are the most "structured" colorings of `C_n`
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— colorings the thin `m = 3` annulus is free to produce, but other
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tires (with larger annulus / heavier chord patterns) actively forbid
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because their interior forces additional colour variety.
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**Consequence.** The simple structural shortcut — "all supports
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contain a common floor, hence pairwise intersections are trivially
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non-empty" — does not hold. Same-`n` compatibility (which by `4CT` is
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still forced to hold) cannot be reduced to a single containment fact;
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it requires a finer structural analysis of *which* colorings sit in
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*which* supports, or a different shortcut entirely.
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@@ -1,72 +1,89 @@
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\relax
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\citation{bauerfeld-nested-tire-duals}
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\citation{birkhoff-reducibility}
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\citation{birkhoff-lewis-chromatic}
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\citation{tutte-chromatic-sums-1973}
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\citation{tutte-algebraic-colorings}
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\citation{tutte-four-colour-conjecture}
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\citation{dvorak-lidicky-cones}
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\citation{heesch-untersuchungen}
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\citation{robertson-sanders-seymour-thomas}
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\citation{robertson-sanders-seymour-thomas}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent }
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\newlabel{def:dual}{{1.3}{2}}
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\newlabel{def:dual-depth}{{1.4}{2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
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\newlabel{fig:dual-depth}{{1}{2}}
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\newlabel{def:dual-component}{{1.5}{2}}
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\newlabel{def:tire-graph}{{1.6}{3}}
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\newlabel{rem:tire-counts}{{1.7}{3}}
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\newlabel{prop:no-level-d-pinch}{{1.8}{3}}
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\newlabel{def:tire-graph}{{1.6}{2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent }
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\newlabel{fig:dual-depth}{{1}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent }
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\newlabel{fig:tire-example}{{2}{4}}
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\newlabel{rem:tire-counts}{{1.7}{4}}
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\newlabel{prop:no-level-d-pinch}{{1.8}{4}}
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\citation{bauerfeld-depth}
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\newlabel{lem:tire-component}{{1.9}{5}}
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\citation{bauerfeld-depth}
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\newlabel{thm:tread-partition}{{1.10}{6}}
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\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
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\newlabel{thm:inner-dual-outerplanar}{{1.13}{7}}
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\newlabel{thm:inner-dual-outerplanar}{{1.13}{8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{8}{}\protected@file@percent }
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\newlabel{fig:inner-dual-disk-case}{{3}{8}}
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\citation{bauerfeld-nested-tire-duals}
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\citation{bauerfeld-nested-tire-duals}
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\citation{tait-original}
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\newlabel{rem:hamilton-cycle-spoke-only}{{1.14}{9}}
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\newlabel{rem:bridge-case-theta}{{1.15}{9}}
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\newlabel{thm:tait-tire}{{1.16}{9}}
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\citation{tait-original}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong \Theta (1, 7, 7)$ is outerplanar.}}{10}{}\protected@file@percent }
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\newlabel{fig:inner-dual-annulus-case}{{4}{10}}
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\newlabel{thm:tait-tire}{{1.16}{10}}
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\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
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\newlabel{def:boundary-state-transfer}{{1.18}{11}}
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\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{11}}
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\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{12}}
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\newlabel{rem:spoke-only-chromatic-transfer}{{1.20}{12}}
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\newlabel{thm:tread-tree}{{1.21}{12}}
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\newlabel{rem:tree-multiple-children}{{1.22}{13}}
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\newlabel{thm:tire-tree-decomposition}{{1.23}{13}}
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\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}}
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\newlabel{thm:tread-tree}{{1.21}{13}}
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\newlabel{rem:tree-multiple-children}{{1.22}{14}}
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\newlabel{thm:tire-tree-decomposition}{{1.23}{14}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.23\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{16}{}\protected@file@percent }
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\newlabel{fig:tire-tree-decomposition}{{5}{16}}
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\newlabel{rem:level-cycle-motivation}{{1.25}{16}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{16}}
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\newlabel{rem:tree-coloring-factorisation}{{1.24}{16}}
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\newlabel{rem:level-cycle-motivation}{{1.25}{17}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{17}}
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\newlabel{conj:false-universal-level-cycle-three-colour}{{1.27}{17}}
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||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $8$-vertex counterexample to the universal-source form. With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and forces all four colours in every proper $4$-vertex-colouring.}}{17}{}\protected@file@percent }
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\newlabel{fig:universal-level-cycle-counterexample}{{6}{17}}
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\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{17}}
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\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{18}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{18}{}\protected@file@percent }
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\newlabel{def:tire-inner-boundary-three-colour}{{1.29}{18}}
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\newlabel{conj:tire-inner-boundary-three-colour}{{1.30}{18}}
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\newlabel{rem:inner-boundary-vs-level-cycle}{{1.31}{18}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{18}{}\protected@file@percent }
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\newlabel{ex:inner-boundary-counterexample}{{1.32}{18}}
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\newlabel{ex:inner-boundary-counterexample}{{1.31}{18}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 1.30\hbox {} in a planar embedding. The six degree-$3$ vertices split into two triples, $\{3,5,10\}$ each adjacent to a triangle in the core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in the core $\{7,8,9,12\}$; the two cores are joined by the edges $17,28,69$ together with $12$.}}{19}{}\protected@file@percent }
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\newlabel{fig:inner-boundary-counterexample}{{7}{19}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{20}{}\protected@file@percent }
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\newlabel{conj:level-cycle-three-colour}{{1.33}{20}}
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||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{19}{}\protected@file@percent }
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||||
\newlabel{conj:level-cycle-three-colour}{{1.32}{19}}
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||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{20}{}\protected@file@percent }
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||||
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{20}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-counts}{{1}{20}}
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||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{20}{}\protected@file@percent }
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||||
\newlabel{def:seam}{{1.34}{20}}
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||||
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the level-cycle three-colour conjecture.}}{21}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{21}}
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\newlabel{def:partial-tire-tree}{{1.35}{21}}
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\newlabel{lem:seam-edge-shared}{{1.36}{21}}
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||||
\newlabel{conj:seam-counterexample}{{1.37}{21}}
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||||
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the level-cycle three-colour conjecture.}}{20}{}\protected@file@percent }
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||||
\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{20}}
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\newlabel{def:seam}{{1.33}{21}}
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\newlabel{def:partial-tire-tree}{{1.34}{21}}
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||||
\newlabel{lem:seam-edge-shared}{{1.35}{21}}
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\newlabel{conj:seam-counterexample}{{1.36}{21}}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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\bibcite{birkhoff-reducibility}{4}
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\bibcite{birkhoff-lewis-chromatic}{5}
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\bibcite{tutte-four-colour-conjecture}{6}
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\bibcite{tutte-algebraic-colorings}{7}
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\bibcite{tutte-chromatic-sums-1973}{8}
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\bibcite{heesch-untersuchungen}{9}
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\bibcite{robertson-sanders-seymour-thomas}{10}
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\bibcite{dvorak-lidicky-cones}{11}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{14.69437pt}
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\newlabel{tocindent1}{17.77782pt}
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@@ -1,5 +1,5 @@
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# Fdb version 3
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["pdflatex"] 1780365355 "paper.tex" "paper.pdf" "paper" 1780365356
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["pdflatex"] 1780415103 "paper.tex" "paper.pdf" "paper" 1780415105
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@@ -147,8 +147,8 @@
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"fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa ""
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"fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d ""
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[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr
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[]
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@@ -82,6 +82,40 @@ companion paper \cite{bauerfeld-nested-tire-duals} uses these
|
||||
definitions to develop nested-cycle structure theorems and
|
||||
chain-pigeonhole conjectures for tire annular subgraphs of $G'$.
|
||||
|
||||
\paragraph{Related work.}
|
||||
The structural object underlying this programme --- the set of
|
||||
proper $4$-colourings of a boundary cycle that extend to a colouring
|
||||
of a bounded planar region --- is classical. Birkhoff's reducibility
|
||||
analysis of the diamond configuration~\cite{birkhoff-reducibility} is
|
||||
the earliest instance of computing such extension sets to attack the
|
||||
Four Colour Theorem; the chromatic polynomial framework of Birkhoff
|
||||
and Lewis~\cite{birkhoff-lewis-chromatic} systematised the counting.
|
||||
Tutte studied how the chromatic polynomial of a rooted planar
|
||||
triangulation decomposes along its outer
|
||||
boundary~\cite{tutte-chromatic-sums-1973} and developed an algebraic
|
||||
theory of graph colourings organised around separating
|
||||
subgraphs~\cite{tutte-algebraic-colorings, tutte-four-colour-conjecture}.
|
||||
The most recent and structurally closest parallel is Dvo\v{r}\'ak
|
||||
and Lidick\'y's analysis of \emph{coloring count
|
||||
cones}~\cite{dvorak-lidicky-cones}, which characterises the possible
|
||||
boundary-extension functions on a fixed outer cycle of a
|
||||
near-triangulation. The Heesch--Appel--Haken
|
||||
approach~\cite{heesch-untersuchungen, robertson-sanders-seymour-thomas}
|
||||
also uses boundary-extension reasoning, but case-by-case on a finite
|
||||
unavoidable set of local configurations rather than as part of a
|
||||
global structural induction.
|
||||
|
||||
The tire-tree decomposition introduced here differs from each of
|
||||
these in shape rather than ingredients. Birkhoff, Tutte, and
|
||||
Dvo\v{r}\'ak--Lidick\'y all study \emph{one} boundary; Heesch and
|
||||
the cleaned-up Appel--Haken proof~\cite{robertson-sanders-seymour-thomas}
|
||||
study a finite collection of local boundaries. The present framework
|
||||
organises the entire triangulation into a hierarchy of annular
|
||||
regions glued along level cycles, and asks whether boundary-extension
|
||||
constraints compose compatibly up the hierarchy. To the authors'
|
||||
knowledge, no prior work on the Four Colour Theorem has been
|
||||
organised around a global nested-cycle decomposition of this kind.
|
||||
|
||||
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
|
||||
with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
|
||||
and $G$ has $2n - 4$ triangular faces.
|
||||
@@ -1373,35 +1407,6 @@ inner-boundary three-colour restriction with respect to
|
||||
$\mathcal{T}(G, \{v_0\})$.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{remark}[Relation to the level-cycle conjecture]
|
||||
\label{rem:inner-boundary-vs-level-cycle}
|
||||
For a depth-$d$ tire $T$, the inner outerplanar graph satisfies
|
||||
$O^{(T)} \subseteq G[L_{d+1}]$: a depth-$d$ face has its three vertex
|
||||
levels in $\{d, d+1\}$ (adjacent vertices differ by at most one level),
|
||||
so the level-$(d+1)$ vertices of the tire's dual component are exactly
|
||||
$V(O^{(T)})$. Since $O^{(T)}$ is outerplanar, every one of its vertices
|
||||
lies on the inner-boundary walk, whence $V(B_{\mathrm{in}}^{(T)}) =
|
||||
V(O^{(T)}) \subseteq L_{d+1}$ is supported on a single level, and is a
|
||||
simple level cycle when $O^{(T)}$ is $2$-connected.
|
||||
|
||||
Consequently the vertex-source form of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} implies
|
||||
Conjecture~\ref{conj:tire-inner-boundary-three-colour} on every
|
||||
$2$-connected inner boundary: the witnessing colouring already makes
|
||||
each such cycle omit a colour. The present conjecture is thus a
|
||||
\emph{weakening}, constraining only the inner-boundary cycles of one
|
||||
tire-tree decomposition rather than all level cycles of some level
|
||||
source. It is no harder than the vertex-source form of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour}, while targeting exactly
|
||||
the interface the chromatic-transfer machinery of
|
||||
Theorem~\ref{thm:tire-chromatic-polynomial-transfer} runs across.
|
||||
(The non-$2$-connected case --- an
|
||||
inner boundary whose walk traverses a bridge or cut-vertex of $O^{(T)}$
|
||||
--- is not covered by the simple-cycle statement of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} and must be argued
|
||||
separately.)
|
||||
\end{remark}
|
||||
|
||||
\subsection*{A counterexample at $n=14$}
|
||||
|
||||
Conjecture~\ref{conj:tire-inner-boundary-three-colour} is in fact
|
||||
@@ -1452,38 +1457,18 @@ $17,28,69$ together with $12$.}
|
||||
The failure was verified by enumerating, for each of the $14$ vertex
|
||||
sources, all $96$ proper $4$-colourings of $G^\star$ and computing the
|
||||
inner boundary $V(B_{\mathrm{in}}^{(T)})$ of every tire $T$ as the
|
||||
level-$(d+1)$ vertices of the corresponding depth-$d$ dual component
|
||||
(Remark~\ref{rem:inner-boundary-vs-level-cycle}). Each source has
|
||||
exactly two non-degenerate inner boundaries (size $\geq 4$), and every
|
||||
proper $4$-colouring assigns all four colours to at least one of them.
|
||||
level-$(d+1)$ vertices of the corresponding depth-$d$ dual component.
|
||||
Each source has exactly two non-degenerate inner boundaries
|
||||
(size $\geq 4$), and every proper $4$-colouring assigns all four
|
||||
colours to at least one of them.
|
||||
|
||||
Because Conjecture~\ref{conj:tire-inner-boundary-three-colour} is a
|
||||
weakening of the vertex-source form of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} only \emph{on
|
||||
$2$-connected inner boundaries}
|
||||
(Remark~\ref{rem:inner-boundary-vs-level-cycle}), $G^\star$ need not
|
||||
refute Conjecture~\ref{conj:level-cycle-three-colour}, and in fact does
|
||||
not: the vertex source $v_0 = 10$ admits a proper $4$-colouring under
|
||||
which every simple level cycle uses at most three colours. Combining
|
||||
this with Remark~\ref{rem:inner-boundary-vs-level-cycle}, at least one
|
||||
tire $T$ under $v_0 = 10$ must have an inner outerplanar graph
|
||||
$O^{(T)}$ that fails to be $2$-connected --- under the witnessing
|
||||
colouring, some inner boundary uses all four colours, and if its
|
||||
$O^{(T)}$ were $2$-connected then by
|
||||
Remark~\ref{rem:inner-boundary-vs-level-cycle} that inner boundary
|
||||
would be a simple level cycle, contradicting the level-cycle witness.
|
||||
The failure of
|
||||
Conjecture~\ref{conj:tire-inner-boundary-three-colour} is therefore
|
||||
attributable to the non-$2$-connected case left open by
|
||||
Remark~\ref{rem:inner-boundary-vs-level-cycle}, not to a deeper failure
|
||||
of the level-cycle statement on $G^\star$.
|
||||
The graph $G^\star$ does not refute
|
||||
Conjecture~\ref{conj:level-cycle-three-colour}: the vertex source
|
||||
$v_0 = 10$ admits a proper $4$-colouring under which every simple level
|
||||
cycle uses at most three colours.
|
||||
|
||||
\subsection*{The surviving level-cycle conjecture}
|
||||
|
||||
The verification on $G^\star$ above is consistent with the broader
|
||||
empirical picture for the level-cycle restriction, which we record as
|
||||
the conjecture this section ultimately stands on.
|
||||
|
||||
\begin{conjecture}[Level-cycle three-colour conjecture]
|
||||
\label{conj:level-cycle-three-colour}
|
||||
Let $G$ be a maximal planar graph. Then there exists a level source
|
||||
@@ -1703,6 +1688,46 @@ E.~Bauerfeld,
|
||||
\emph{Coloring Nested Tire Dual Graphs},
|
||||
manuscript (math-research repository), 2026.
|
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|
||||
\bibitem{birkhoff-reducibility}
|
||||
G.~D.~Birkhoff,
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||||
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|
||||
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\bibitem{birkhoff-lewis-chromatic}
|
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G.~D.~Birkhoff and D.~C.~Lewis,
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\emph{Chromatic polynomials},
|
||||
Trans.\ Amer.\ Math.\ Soc.\ \textbf{60} (1946), 355--451.
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\bibitem{tutte-four-colour-conjecture}
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W.~T.~Tutte,
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\emph{On the four-colour conjecture},
|
||||
Proc.\ London Math.\ Soc.\ (2) \textbf{50} (1948), 137--149.
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|
||||
\bibitem{tutte-algebraic-colorings}
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|
||||
\bibitem{tutte-chromatic-sums-1973}
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W.~T.~Tutte,
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Canad.\ J.\ Math.\ \textbf{25} (1973), 426--447.
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|
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\bibitem{heesch-untersuchungen}
|
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H.~Heesch,
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|
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\bibitem{robertson-sanders-seymour-thomas}
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N.~Robertson, D.~P.~Sanders, P.~D.~Seymour, and R.~Thomas,
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\bibitem{dvorak-lidicky-cones}
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Z.~Dvo\v{r}\'ak and B.~Lidick\'y,
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|
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J.\ Graph Theory \textbf{100} (2022), 84--100.
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|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user