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\title{Birkhoff--Heesch reducibility and the fiber view\\
       \large A dictionary between classical 4CT reducibility theory
       and our spoke-fiber decomposition}
\author{}
\date{}

\newtheorem*{prop}{Proposition}
\newtheorem*{thm}{Theorem}
\newtheorem*{defn}{Definition}

\begin{document}
\maketitle

\section*{Purpose}

The conversation around the fiber decomposition note
(\texttt{fiber\_decomposition.tex}) flagged that the technique of
``split a coloring into a boundary configuration plus an
extension-counting fiber'' is exactly the framework of classical
reducibility theory for the Four-Color Theorem (4CT) --- developed by
Birkhoff (1913), then Heesch, Bernhart, Allaire, Swart, Appel--Haken,
and finally Robertson--Sanders--Seymour--Thomas (RSST, 1997).  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}
