diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.aux b/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.aux new file mode 100644 index 0000000..a0fa2b6 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.aux @@ -0,0 +1,13 @@ +\relax +\citation{thomas-update} +\citation{thomas-update} +\citation{bauerfeld-pds} +\citation{tutte-chromial} +\bibcite{birkhoff}{1} +\bibcite{heesch}{2} +\bibcite{appel-haken}{3} +\bibcite{rsst}{4} +\bibcite{thomas-update}{5} +\bibcite{tutte-chromial}{6} +\bibcite{bauerfeld-pds}{7} +\gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.log b/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.log new file mode 100644 index 0000000..bcae532 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.log @@ -0,0 +1,312 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 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This +note (a) summarises the relevant pieces of that machinery in modern +notation following Thomas's survey~\cite{thomas-update}, then (b) +gives an explicit dictionary to our spoke-fiber language and an +honest assessment of what carries over. + +\section*{Part I: classical reducibility, in modern notation} + +\subsection*{The 4CT and its Tait dual} + +\begin{thm}[4CT, vertex form] +Every plane graph $G$ has a proper $4$-vertex-coloring. +\end{thm} + +\begin{thm}[Tait 1880, edge form] +Every cubic plane graph $H$ with no cut-edge has a proper $3$-edge-coloring. +This is equivalent to the 4CT. +\end{thm} + +The equivalence goes through planar duality: a $4$-coloring of a +triangulation $G$, with colors $(0,0), (1,0), (0,1), (1,1)$ in +$\mathbb{Z}_2 \times \mathbb{Z}_2$, induces a $3$-edge-coloring of the +dual $G^*$ by assigning each dual edge the sum of the colors of the +two regions it separates. This is exactly the side our paper works +on: a triangulation $G$ has a cubic dual $G' \subseteq G^*$ +(specifically: $G^*$ minus the unbounded face), and edge $3$-colorings +of $G'$ \emph{are} (up to handling of the outer face) $4$-colorings of $G$. + +\subsection*{Birkhoff's minimum-counterexample setup (1913)} + +The reducibility programme assumes for contradiction that there is a +\emph{minimum counterexample} to the 4CT --- a plane graph $T$ that is +not $4$-colorable but every smaller plane graph is. Birkhoff showed: + +\begin{thm}[Birkhoff 1913] +Every minimum counterexample to the 4CT is an internally +$6$-connected triangulation. +\end{thm} + +(``Internally $6$-connected'' means: removing fewer than $5$ vertices +keeps the graph connected, and removing any $5$ vertices leaves at +most a single isolated vertex.) All subsequent work assumes $T$ has +this form. + +\subsection*{Configurations, rings, free completions} + +A \emph{configuration} captures a small local piece of $T$ +together with degree information. + +\begin{defn}[Configuration, Thomas~\cite{thomas-update}] +A configuration is a pair $K = (H, \gamma)$ where $H$ is a +near-triangulation (one face is designated as ``special'', and every +other face is a triangle) and $\gamma : V(H) \to \mathbb{Z}_{\geq 5}$ +satisfies: +\begin{enumerate} + \item[(i)] for interior vertices $v$ (not on the special face), + $\gamma(v) = \deg_H(v)$; + \item[(ii)] for boundary vertices $v$ (on the special face), + $\gamma(v) > \deg_H(v)$; + \item[(iii)] $\operatorname{ring-size}(K) + := \sum_{v \text{ boundary}, H \setminus v \text{ connected}} + (\gamma(v) - \deg_H(v) - 1) \geq 2$. +\end{enumerate} +$K$ \emph{appears} in a triangulation $T$ if $H$ is an induced subgraph +of $T$, every non-special face of $H$ is a face of $T$, and $\gamma(v)$ +equals the degree of $v$ in $T$ for every $v \in V(H)$. +\end{defn} + +\begin{defn}[Free completion] +The \emph{free completion} of $K$ is the (essentially unique) +plane graph $S$ obtained from $H$ by adding a single cycle $R$ --- the +\emph{ring} of $K$ --- around the special face, plus the unique set of +edges making $S$ a triangulation of the disk bounded by $R$ in which +every vertex of $H$ has degree exactly $\gamma$. $R$ has length equal +to the ring-size of $K$. +\end{defn} + +\subsection*{Color sets and reducibility} + +Let $\mathcal{K}(R)$ be the set of all proper $4$-colorings of the ring +$R$ (often considered up to the $S_4$-action on colors, leaving roughly +$|\mathcal{K}(R)|/24$ orbits). + +\begin{defn}[Good and bad colorings] +A coloring $\varphi \in \mathcal{K}(R)$ is \emph{good} if it extends +to a proper $4$-coloring of the free completion $S$, i.e.\ to all of +$H \cup R$. Otherwise it is \emph{bad}. Write +$\mathcal{C} \subseteq \mathcal{K}(R)$ for the set of good colorings. +\end{defn} + +Now suppose $K$ appears in a minimum counterexample $T$, with free +completion $S \subseteq T$. Let $T' := T \setminus V(H)$ (the +``outside''); since $T$ is a minimum counterexample, $T'$ has a +$4$-coloring, which restricts to some +$\mathcal{C}' \subseteq \mathcal{K}(R)$. For $T$ to be a +counterexample, +\[ + \mathcal{C}' \;\subseteq\; \mathcal{K}(R) \setminus \mathcal{C} +\] +(otherwise a coloring extending both into $S$ and into $T'$ gives a +$4$-coloring of all of $T$). The goal of reducibility is to show +$\mathcal{C}' = \emptyset$, contradicting that $T'$ is colorable. + +\begin{defn}[Reducibility, classical taxonomy] +\hfill +\begin{itemize} + \item[A] $K$ is \textbf{A-reducible} if $\mathcal{C} = \mathcal{K}(R)$ + (every ring coloring extends). A-reducibility immediately gives + $\mathcal{C}' = \emptyset$, but is too strong to hold for any + nontrivial configuration. + \item[D] $K$ is \textbf{D-reducible} if every bad coloring + $\varphi \in \mathcal{K}(R) \setminus \mathcal{C}$ can be + transformed --- via a sequence of Kempe-chain swaps on $T'$ --- + into a good coloring. D-reducibility is checkable by computer + on $\mathcal{K}(R)$ alone (no knowledge of $T'$ beyond that it + is $4$-colorable), because the Kempe-swap closure operation + can be applied at the level of ring colorings. + \item[C] $K$ is \textbf{C-reducible} if it is D-reducible \emph{after} + replacing $H$ by a smaller graph $H'$ (obtained from $H$ by + contracting up to four edges in RSST). This is strictly more + powerful than D-reducibility. +\end{itemize} +\end{defn} + +Birkhoff's diamond (4 mutually adjacent pentagons surrounded by a +$6$-cycle ring) was the first nontrivial D-reducible configuration: +of the 31 ring colorings up to $S_4$-symmetry, 16 are good directly, +and the other 15 each admit a Kempe-swap to a good one. + +\subsection*{Discharging and unavoidability} + +Reducibility tells you that \emph{if} a good configuration appears in +$T$, then $T$ is not a counterexample. The other half of the proof +is to show that some good configuration \emph{must} appear: + +\begin{defn}[Unavoidable set] +A set $\mathcal{U}$ of configurations is \emph{unavoidable} if every +internally $6$-connected triangulation contains at least one $K \in +\mathcal{U}$ as a subconfiguration. +\end{defn} + +Heesch's \emph{discharging method} proves unavoidability: +\begin{enumerate} + \item Assign a \emph{charge} $\operatorname{ch}(v) := 6 - \deg(v)$ to each + vertex. By Euler's formula, $\sum_v \operatorname{ch}(v) = 12$ on + any triangulation of the sphere. + \item Define \emph{discharging rules} that redistribute charge + between vertices without changing the total. + \item Show that after discharging, no vertex carries positive charge + \emph{unless} it lies in (the neighborhood of) some + configuration in $\mathcal{U}$. + \item Since total charge is $12 > 0$, some positive-charge vertex + exists post-discharging, so some configuration in + $\mathcal{U}$ must appear. +\end{enumerate} + +The two halves combine: 4CT $=$ ``every reducible configuration in +$\mathcal{U}$ blocks counterexamples'' $+$ ``$\mathcal{U}$ is unavoidable.'' + +\subsection*{The proofs} + +\begin{itemize} + \item \textbf{Appel--Haken (1976/77).} $|\mathcal{U}| = 1936$ + configurations (later reduced to $1482$); $487$ discharging + rules. Computer verification needed. + \item \textbf{RSST (1997).} $|\mathcal{U}| = 633$ configurations; $32$ + discharging rules; quadratic-time $4$-coloring algorithm. + Still computer-assisted, but the unavoidability part was + written in a formal language and machine-checked. +\end{itemize} + +For configurations of ring-size $r$, the number of colorings of $R$ +modulo $S_4$ is roughly $3^{r-1}/3 + O(2^r)$ from the standard +chromatic-polynomial formula for $C_r$. RSST's largest ring-size is +$14$, with $\sim\!200{,}000$ ring colorings per configuration to +check. + +\section*{Part II: dictionary to the fiber-decomposition view} + +\subsection*{Side-of-the-duality conventions} + +Our work is on the \emph{edge} side of Tait: we color edges of $G'$ +with $3$ colors, where $G'$ is essentially the dual of a triangulation +$G$. Birkhoff--Heesch is on the \emph{vertex} side of $G$. By Tait's +theorem these are equivalent in principle, but the shape of a +``configuration boundary'' looks different on the two sides: +\begin{center} +\begin{tabular}{>{\raggedright\arraybackslash}p{0.36\textwidth}|>{\raggedright\arraybackslash}p{0.55\textwidth}} +\textbf{Vertex side (Birkhoff--Heesch)} & \textbf{Edge side (us)} \\ \hline +Triangulation $T$ & Cubic dual $G'$ \\ +Configuration $K = (H, \gamma)$ in $T$ & Tire annular face connector + $T'_{f'} \subseteq G'$ \\ +Inner-face triangles of $H$ & Vertices of $V(f')$ (dual to annular + faces of the tire) \\ +Ring $R$ (a cycle around $H$) & Boundary of $T'_{f'}$, consisting of + spoke edges $E_S$ \emph{plus} (in the multi-tire chain) the cycle + $V(f')$ itself \\ +Ring coloring $\varphi : V(R) \to \{1,2,3,4\}$ & Spoke configuration + $\sigma : E_S \to \{1,2,3\}$ \\ +$\mathcal{K}(R)$ = all proper $4$-colorings of $R$ & + $\Sigma := $ all spoke configurations $\sigma$ (whether + realisable or not) \\ +$\mathcal{C}$ = good (extendable to $H \cup R$) ring colorings & + Realisable spoke configurations: + $\{\sigma : N(T'_{f'}; \sigma) > 0\}$ \\ +$\mathcal{K}(R) \setminus \mathcal{C}$ = bad ring colorings & + Unrealisable $\sigma$: the boundary of $T'_{f'}$'s reachability \\ +Fiber over a ring coloring: \# extensions of $\varphi$ to $H \cup R$ & + Fiber count $N(T'_{f'}; \sigma)$ \\ +Kempe-chain swap on $T \setminus V(H)$ & + Tait-Kempe swap on $G' \setminus V(T'_{f'})$ (two-color + alternating-edge swap) +\end{tabular} +\end{center} + +\subsection*{The fiber identity, in Birkhoff language} + +Our identity +$P_e(T'_{f'}, 3) = \sum_\sigma N(T'_{f'}; \sigma)$ +is literally Birkhoff's ``good colorings'' decomposition on the edge +side, with \emph{the same scaffold}: count interior extensions +fibered over boundary state. The classical literature does not +emphasise this identity per se (it's the trivial sum), but every +reducibility analysis is built around the fiber distribution +$\{N(T'_{f'}; \sigma)\}_\sigma$. + +\subsection*{What ``reducible'' means in our world} + +Translating definitions of A-, D-, C-reducibility through the +dictionary: + +\begin{itemize} + \item \textbf{Edge-A-reducibility of $T'_{f'}$}: every spoke + configuration $\sigma \in \Sigma$ is realisable. This would + mean $T'_{f'}$ supports an edge $3$-coloring extension from + \emph{any} boundary input --- a very strong locally-flexible + property. + \item \textbf{Edge-D-reducibility of $T'_{f'}$}: every unrealisable + $\sigma$ can be Tait-Kempe-swapped (in the outside graph + $G' \setminus T'_{f'}$) to a realisable one. This is the + natural form of the conductivity step from our chain-pigeonhole + sketch. + \item \textbf{Edge-C-reducibility of $T'_{f'}$}: D-reducibility + after replacing $T'_{f'}$ by a smaller subgraph (e.g.\ + contracting some spoke edges or merging adjacent faces). +\end{itemize} + +\subsection*{The chain-pigeonhole step as a reducibility statement} + +Our conductivity step --- ``the middle tire $T_B$'s +$\phi_B : \mathcal{P}_{AB} \to 2^{\mathcal{P}_{BC}}$ takes any input +to a $>\!|\mathcal{P}_{BC}|/2$-sized output'' --- is the +\emph{quantitative} analogue of D-reducibility, but applied to +\emph{composition} of configurations rather than reduction. In +particular: + +\begin{itemize} + \item Birkhoff--Heesch D-reducibility says ``bad inputs to a single + configuration can be Kempe-moved to good ones.'' + \item Our chain-pigeonhole says ``the bad inputs to the inner tire + and the bad inputs to the outer tire together don't cover + $\mathcal{P}_\gamma$, so something good remains.'' +\end{itemize} + +These are not the same statement, but they are about the same data +$\mathcal{C}$ vs.\ $\mathcal{K} \setminus \mathcal{C}$ on the shared +cycle. + +\section*{Part III: does the machinery apply?} + +\subsection*{What carries over cleanly} + +\begin{enumerate} + \item \textbf{Vocabulary and scaffolding.} Configurations, + rings, free completions, color sets, fiber-of-extensions are + all there; the menagerie/fiber notes are using the same + objects. + \item \textbf{Kempe-chain machinery.} Tait-Kempe chains (two-color + alternating edge paths in $G'$) are a real, well-developed + tool. Any conductivity argument we want is fundamentally + Tait-Kempe in flavour. + \item \textbf{Computer-verifiability for small tires.} Just as + D-reducibility of an individual configuration is verified by + enumerating $\sim\!200{,}000$ ring colorings, the fiber + distribution $\{N(T'_{f'}; \sigma)\}$ for a fixed small tire + is a finite Sage computation. We can test conductivity + empirically before trying to prove it. +\end{enumerate} + +\subsection*{What does \emph{not} carry over straightforwardly} + +\begin{enumerate} + \item \textbf{The reducibility scale.} Classical reducibility was + only ever practical for ring-size $\leq 14$ because the number + of ring colorings explodes. Our tires can have arbitrarily + large boundary cycles (in a multi-layer triangulation, the + annular ring is whatever the level structure gives), so the + \emph{single-configuration} reducibility approach hits the + same wall as A\&H/RSST: only small tires are tractable + directly. + \item \textbf{Compositionality.} Birkhoff--Heesch operates on + \emph{one} configuration at a time. Our chain + pigeonhole/nesting is fundamentally about \emph{composing} + configurations along shared boundaries. This is a structural + feature classical reducibility does \emph{not} engage with --- + their unavoidability argument (discharging) replaces it. If + we want to prove a statement of the form ``every nested chain + of tires admits a global $4$-coloring,'' Birkhoff--Heesch does + not directly give us the tool; we need either a transfer-matrix + / monotonicity argument across nestings or a structural + result about how the realisable supports behave under + composition. + \item \textbf{Unavoidability is automatic / different.} In 4CT, + unavoidability is a separate hard problem solved by + discharging. In our setup the tire decomposition of + $G$ is given by the level structure (Bauerfeld + \cite{bauerfeld-pds}), so there is no analogous unavoidability + question --- the tires are already there. This means + \emph{half} of the classical apparatus (discharging) is not + the point of contact; the contact is purely on the + reducibility/color-set side. + \item \textbf{Tait correspondence is global, not local.} Edge + $3$-coloring of $G'$ globally encodes a vertex $4$-coloring of + $G$, but the local correspondence between an edge coloring of + $T'_{f'}$ and a vertex coloring of the corresponding piece of + $G$ is subtle: edge swaps on the dual side do not always + correspond to single Kempe chains on the primal side, and the + ``ring'' as a vertex cycle in $G$ may differ in length and + structure from the ``boundary'' of $T'_{f'}$ on the edge side. + Any time we want to import a vertex-side Kempe argument we + will have to do the bookkeeping carefully. +\end{enumerate} + +\subsection*{Assessment} + +\textbf{Yes, the Birkhoff--Heesch framework applies, but only as a +language and as a verification tool for small instances.} It does +\emph{not} hand us a proof of the nesting/chain-pigeonhole conjecture +for free, because: +\begin{itemize} + \item Classical reducibility is single-configuration; our argument + is multi-configuration / compositional. + \item Classical reducibility's quantitative input is exhaustive + enumeration of ring colorings up to size $14$; we want + statements uniform over arbitrarily large tires. +\end{itemize} + +What it \emph{does} hand us: +\begin{itemize} + \item A precise vocabulary --- good/bad colorings, free completion, + D-/C-reducibility --- so that the conductivity step can be + stated in established terms. + \item A concrete computational template: for any specific tire, + compute $\{N(T'_{f'}; \sigma)\}_\sigma$ in Sage and check + whether the realisable support $\mathcal{C}$ is large enough + that two adjacent tires must overlap. This is the + analogue of mechanical D-reducibility checking. + \item Strong evidence about the difficulty: nothing in 80+ years of + reducibility work has reduced 4CT to a structural argument + across all configuration sizes. If our chain argument + succeeds, it will be \emph{because tires are a more + structured class than arbitrary configurations}, not because + the reducibility apparatus suddenly gives uniform results. +\end{itemize} + +\subsection*{Concrete next steps} + +\begin{enumerate} + \item \textbf{Pick a small tire family} ($B_{\mathrm{in}}$ a $k$-cycle + for $k \in \{3, 4, 5, 6\}$, $B_{\mathrm{out}}$ a small cycle, + no $O$-chords) and compute the full fiber distribution + $\{N(T'_{f'}; \sigma)\}_\sigma$ in Sage. This gives the realisable + support $\mathcal{C}$ and, in classical terms, tests whether + small tires are A-reducible (full support), D-reducible + (Kempe-recoverable), or neither. + \item \textbf{Check support overlap for adjacent tires.} Given two + small tires sharing a cycle $\gamma$, do the realisable + supports $\mathcal{C}_{\mathrm{out}}, \mathcal{C}_{\mathrm{in}} + \subseteq \mathcal{K}(\gamma)$ always intersect? This is the + empirical version of the chain-pigeonhole step. If they + sometimes miss, the simple form of the argument fails and + we need a Kempe-chain (D-reducibility-style) escape route. + \item \textbf{Look up monosystems / Tutte.} Tutte + \cite{tutte-chromial} formulated 4CT-adjacent counting + problems in terms of his \emph{chromial}; this is the closest + existing transfer-matrix-style framing of these color sets, + and may give a cleaner composition rule than the raw fiber + sum. + \item \textbf{Look up Sokal / Chang--Shrock} on chromatic + polynomials of strip graphs. They use transfer matrices for + infinite families of small-ring configurations, which is + structurally similar to nested tires with fixed ring size. +\end{enumerate} + +\begin{thebibliography}{9} + +\bibitem{birkhoff} +G.~D.~Birkhoff, +\emph{The reducibility of maps}, +Amer.\ J.\ Math.\ \textbf{35} (1913), 115--128. + +\bibitem{heesch} +H.~Heesch, +\emph{Untersuchungen zum Vierfarbenproblem}, +B.~I.\ Hochschulskripten 810/810a/810b, Bibliographisches Institut, +Mannheim, 1969. + +\bibitem{appel-haken} +K.~Appel and W.~Haken, +\emph{Every planar map is four colorable. Part I: Discharging}; +\emph{Part II: Reducibility} (with J.~Koch), +Illinois J.\ Math.\ \textbf{21} (1977), 429--567. + +\bibitem{rsst} +N.~Robertson, D.~P.~Sanders, P.~Seymour, R.~Thomas, +\emph{The Four-Colour Theorem}, +J.\ Combin.\ Theory Ser.\ B \textbf{70} (1997), 2--44. + +\bibitem{thomas-update} +R.~Thomas, +\emph{An update on the Four-Color Theorem}, +Notices Amer.\ Math.\ Soc.\ \textbf{45} (1998), 848--859. + +\bibitem{tutte-chromial} +W.~T.~Tutte, +\emph{Chromials}, +in: \emph{Hypergraph Seminar} (C.~Berge, D.~Ray-Chaudhuri, eds.), +Lecture Notes in Math.\ 411, Springer, 1974, pp.~243--266. + +\bibitem{bauerfeld-pds} +E.~Bauerfeld, +\emph{Plane Depth Sequencing}, +manuscript (math-research repository), 2026. + +\end{thebibliography} + +\end{document}