diff --git a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py b/papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py index 3f567d8..6836b11 100644 --- a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py +++ b/papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py @@ -327,6 +327,38 @@ def enumerate_level_cycle_supports( return supports, stats +def check_floor_containment( + supports: dict[frozenset[tuple[int, ...]], list[dict]], +) -> None: + """Test whether the smallest-size support is a subset of every other support. + + The supports are fully normalized (orbits under S_4 x D_n), so set + inclusion here is the right test for the "floor support is contained in + every admissible support" conjecture. + """ + if len(supports) < 2: + print(" containment: <2 supports, trivially contained") + return + keys = list(supports.keys()) + min_size = min(len(k) for k in keys) + floors = [k for k in keys if len(k) == min_size] + print(f" containment check: floor support (size {min_size}) vs {len(keys) - 1} others") + for floor in floors: + bad: list[tuple[int, tuple[int, ...]]] = [] + for other in keys: + if other is floor: + continue + if not floor.issubset(other): + missing = next(iter(floor - other)) + bad.append((len(other), missing)) + if not bad: + print(f" floor IS a subset of all {len(keys) - 1} other supports") + else: + print(f" floor is NOT a subset of {len(bad)} of {len(keys) - 1} other supports") + for size, missing in bad[:5]: + print(f" missing example: floor coloring {missing} absent from support of size {size}") + + def summarize( n: int, supports: dict[frozenset[tuple[int, ...]], list[dict]], @@ -357,6 +389,7 @@ def summarize( print(f" most restrictive support : {min_size}/{all_colorings} ({len(min_supports)} support types)") print(f" least restrictive support : {max_size}/{all_colorings}") print(f" overlap among max-restrict : min {min(pair_overlaps)}, max {max(pair_overlaps)}") + check_floor_containment(supports) print(" examples:") printed = 0 diff --git a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md b/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md index e54e9fe..09410c6 100644 --- a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md +++ b/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md @@ -239,3 +239,73 @@ opens the door to closed-form support computation for the worst case. The obstruction to a quick proof is exotic `m ≥ 4` configurations not reachable by subdivision from an `m = 3` floor witness — none has appeared empirically, but their non-existence is not yet ruled out. + +#### Skip-`m=3` sweep (partial) + +To probe the conjecture more directly we ran a "skip-`m=3`" sweep +explicitly excluding `m = 3` from the search: + +```bash +python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \ + --n-min 6 --n-max 6 --outer-min 4 --outer-max 10 --inner-min 7 --inner-max 9 \ + --progress +``` + +The sweep ran for `11h 49m` and was terminated partway through `m = 8, +k = 9` at chord-set #6 of the 2-chord block (raw=6.45M canonical=223k). +Throughout the run --- across `m ∈ [4, 7]` exhaustively and `m = 8` +partially --- the distinct-support count stayed locked at **`13`** +(vs `19` for the full `m ∈ [3, 7]` run). The six missing support +types are precisely the ones with `m = 3`-only witnesses, including +the unique `252/732` floor. + +Conclusion of the partial run: no support of size `≤ 252` appeared +under `m ≥ 4` in any configuration searched, and the support count +saturated early, suggesting the support landscape for `m ≥ 4` is +strictly looser than for `m = 3` even as `m` grows. The conjecture is +not falsified by any data we have; pushing `m` to `9` or `10` is +left as future work (path-count growth makes it expensive: `m = 10, +k = 9` alone has `C(19, 10) ≈ 92{,}000` paths per chord set). + +### Floor-containment conjecture — refuted + +Floor-stability bounds the *size* of the worst-case support but not +its content. For the tire-tree compatibility argument, what would +have closed the loop is a containment statement: + +> **(Conjecture, refuted)** For every `n`, the unique floor support +> `F_n` is a *subset* of every admissible level-cycle support. Every +> level-cycle coloring that the `m = 3` floor witness allows is also +> allowed by every other tire. + +The `check_floor_containment` helper in `level_cycle_support.py` +tests this directly on the computed supports. The conjecture **fails +at `n = 4` and `n = 6`**: + +```text +n=4: floor (size 60) is NOT a subset of 1 of 2 other supports + missing: floor coloring (1, 2, 1, 2) absent from support of size 72 + +n=6: floor (size 252) is NOT a subset of 11 of 16 other supports + missing: floor coloring (1, 3, 2, 1, 3, 2) absent from support of size 708 + missing: floor coloring (2, 3, 2, 1, 3, 1) absent from support of size 492 + missing: floor coloring (2, 3, 2, 3, 2, 3) absent from support of size 720 +``` + +(`n = 3` is trivial — only one support type. `n = 5` happens to +satisfy containment in the searched window but only against a single +other support, so the data is too thin to be evidence either way.) + +The missing colorings cluster on **bipartite alternations** +(`(1,2,1,2)`, `(2,3,2,3,2,3)`) and **3-periodic patterns** +(`(1,3,2,1,3,2)`). These are the most "structured" colorings of `C_n` +— colorings the thin `m = 3` annulus is free to produce, but other +tires (with larger annulus / heavier chord patterns) actively forbid +because their interior forces additional colour variety. + +**Consequence.** The simple structural shortcut — "all supports +contain a common floor, hence pairwise intersections are trivially +non-empty" — does not hold. Same-`n` compatibility (which by `4CT` is +still forced to hold) cannot be reduced to a single containment fact; +it requires a finer structural analysis of *which* colorings sit in +*which* supports, or a different shortcut entirely. diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 9c6133f..a7f14af 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -1,72 +1,89 @@ \relax \citation{bauerfeld-nested-tire-duals} +\citation{birkhoff-reducibility} +\citation{birkhoff-lewis-chromatic} +\citation{tutte-chromatic-sums-1973} +\citation{tutte-algebraic-colorings} +\citation{tutte-four-colour-conjecture} +\citation{dvorak-lidicky-cones} +\citation{heesch-untersuchungen} +\citation{robertson-sanders-seymour-thomas} +\citation{robertson-sanders-seymour-thomas} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } \newlabel{def:dual}{{1.3}{2}} \newlabel{def:dual-depth}{{1.4}{2}} -\@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 } -\newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:dual-component}{{1.5}{2}} -\newlabel{def:tire-graph}{{1.6}{3}} -\newlabel{rem:tire-counts}{{1.7}{3}} -\newlabel{prop:no-level-d-pinch}{{1.8}{3}} +\newlabel{def:tire-graph}{{1.6}{2}} +\@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 } +\newlabel{fig:dual-depth}{{1}{3}} \@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 } \newlabel{fig:tire-example}{{2}{4}} +\newlabel{rem:tire-counts}{{1.7}{4}} +\newlabel{prop:no-level-d-pinch}{{1.8}{4}} \citation{bauerfeld-depth} \newlabel{lem:tire-component}{{1.9}{5}} \citation{bauerfeld-depth} \newlabel{thm:tread-partition}{{1.10}{6}} \newlabel{rem:tire-component-degenerate}{{1.11}{7}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} -\newlabel{thm:inner-dual-outerplanar}{{1.13}{7}} +\newlabel{thm:inner-dual-outerplanar}{{1.13}{8}} \@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 } \newlabel{fig:inner-dual-disk-case}{{3}{8}} \citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals} -\citation{tait-original} \newlabel{rem:hamilton-cycle-spoke-only}{{1.14}{9}} \newlabel{rem:bridge-case-theta}{{1.15}{9}} -\newlabel{thm:tait-tire}{{1.16}{9}} +\citation{tait-original} \@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 } \newlabel{fig:inner-dual-annulus-case}{{4}{10}} +\newlabel{thm:tait-tire}{{1.16}{10}} \newlabel{rem:count-general-outerplanar}{{1.17}{11}} \newlabel{def:boundary-state-transfer}{{1.18}{11}} -\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{11}} +\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{12}} \newlabel{rem:spoke-only-chromatic-transfer}{{1.20}{12}} -\newlabel{thm:tread-tree}{{1.21}{12}} -\newlabel{rem:tree-multiple-children}{{1.22}{13}} -\newlabel{thm:tire-tree-decomposition}{{1.23}{13}} -\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}} +\newlabel{thm:tread-tree}{{1.21}{13}} +\newlabel{rem:tree-multiple-children}{{1.22}{14}} +\newlabel{thm:tire-tree-decomposition}{{1.23}{14}} \@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 } \newlabel{fig:tire-tree-decomposition}{{5}{16}} -\newlabel{rem:level-cycle-motivation}{{1.25}{16}} -\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{16}} +\newlabel{rem:tree-coloring-factorisation}{{1.24}{16}} +\newlabel{rem:level-cycle-motivation}{{1.25}{17}} +\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{17}} \newlabel{conj:false-universal-level-cycle-three-colour}{{1.27}{17}} \@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 } \newlabel{fig:universal-level-cycle-counterexample}{{6}{17}} -\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{17}} +\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{18}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{18}{}\protected@file@percent } \newlabel{def:tire-inner-boundary-three-colour}{{1.29}{18}} \newlabel{conj:tire-inner-boundary-three-colour}{{1.30}{18}} -\newlabel{rem:inner-boundary-vs-level-cycle}{{1.31}{18}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{18}{}\protected@file@percent } -\newlabel{ex:inner-boundary-counterexample}{{1.32}{18}} +\newlabel{ex:inner-boundary-counterexample}{{1.31}{18}} \@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 } \newlabel{fig:inner-boundary-counterexample}{{7}{19}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{20}{}\protected@file@percent } -\newlabel{conj:level-cycle-three-colour}{{1.33}{20}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{19}{}\protected@file@percent } +\newlabel{conj:level-cycle-three-colour}{{1.32}{19}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{20}{}\protected@file@percent } \@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 } \newlabel{tab:level-cycle-three-colour-counts}{{1}{20}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{20}{}\protected@file@percent } -\newlabel{def:seam}{{1.34}{20}} -\@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 } -\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{21}} -\newlabel{def:partial-tire-tree}{{1.35}{21}} -\newlabel{lem:seam-edge-shared}{{1.36}{21}} -\newlabel{conj:seam-counterexample}{{1.37}{21}} +\@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 } +\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{20}} +\newlabel{def:seam}{{1.33}{21}} +\newlabel{def:partial-tire-tree}{{1.34}{21}} +\newlabel{lem:seam-edge-shared}{{1.35}{21}} +\newlabel{conj:seam-counterexample}{{1.36}{21}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} +\bibcite{birkhoff-reducibility}{4} +\bibcite{birkhoff-lewis-chromatic}{5} +\bibcite{tutte-four-colour-conjecture}{6} +\bibcite{tutte-algebraic-colorings}{7} +\bibcite{tutte-chromatic-sums-1973}{8} +\bibcite{heesch-untersuchungen}{9} +\bibcite{robertson-sanders-seymour-thomas}{10} +\bibcite{dvorak-lidicky-cones}{11} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index 7bf8e49..10d5def 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780365355 "paper.tex" "paper.pdf" "paper" 1780365356 +["pdflatex"] 1780415103 "paper.tex" "paper.pdf" "paper" 1780415105 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -147,8 +147,8 @@ "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa "" "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d "" "fig_universal_level_cycle_counterexample.png" 1780325973 75145 08f600be4e05c11d702bee45996ca222 "" - "paper.aux" 1780365356 8877 506e0c8fb0cf3332db2191566c33d175 "pdflatex" - "paper.tex" 1780365326 81811 cdae7a96d08d9776f308c7c25eefa561 "" + "paper.aux" 1780415105 9575 5bc3b6429f986d3a3f92f9358d9c8487 "pdflatex" + "paper.tex" 1780414965 82330 193cfffcd3ef710f6995d7f672780ac4 "" (generated) "paper.aux" "paper.log" diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index c5d55f5..202edf0 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 1 JUN 2026 21:55 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 2 JUN 2026 11:45 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -495,107 +495,96 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - + File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 131. +Package pdftex.def Info: fig_dual_depth.png used on input line 165. 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PDF statistics: 220 PDF objects out of 1000 (max. 8388607) 132 compressed objects within 2 object streams diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index aa523c3..1e6e88d 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index cce18bc..4f8b168 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -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. +\bibitem{birkhoff-reducibility} +G.~D.~Birkhoff, +\emph{The reducibility of maps}, +Amer.\ J.\ Math.\ \textbf{35} (1913), 115--128. + +\bibitem{birkhoff-lewis-chromatic} +G.~D.~Birkhoff and D.~C.~Lewis, +\emph{Chromatic polynomials}, +Trans.\ Amer.\ Math.\ Soc.\ \textbf{60} (1946), 355--451. + +\bibitem{tutte-four-colour-conjecture} +W.~T.~Tutte, +\emph{On the four-colour conjecture}, +Proc.\ London Math.\ Soc.\ (2) \textbf{50} (1948), 137--149. + +\bibitem{tutte-algebraic-colorings} +W.~T.~Tutte, +\emph{On the algebraic theory of graph colorings}, +J.\ Combin.\ Theory \textbf{1} (1966), 15--50. + +\bibitem{tutte-chromatic-sums-1973} +W.~T.~Tutte, +\emph{Chromatic sums for rooted planar triangulations: the cases $\lambda = 1$ and $\lambda = 2$}, +Canad.\ J.\ Math.\ \textbf{25} (1973), 426--447. + +\bibitem{heesch-untersuchungen} +H.~Heesch, +\emph{Untersuchungen zum Vierfarbenproblem}, +Hochschulskriptum 810/a/b, Bibliographisches Institut, Mannheim, 1969. + +\bibitem{robertson-sanders-seymour-thomas} +N.~Robertson, D.~P.~Sanders, P.~D.~Seymour, and R.~Thomas, +\emph{The four-colour theorem}, +J.\ Combin.\ Theory Ser.\ B \textbf{70} (1997), 2--44. + +\bibitem{dvorak-lidicky-cones} +Z.~Dvo\v{r}\'ak and B.~Lidick\'y, +\emph{Coloring count cones of planar graphs}, +J.\ Graph Theory \textbf{100} (2022), 84--100. + \end{thebibliography} \end{document}