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\title{Fiber distributions for tire face connectors with chords in $O$}
\author{}
\date{}

\newtheorem*{obs}{Observation}

\begin{document}
\maketitle

\section*{What this is}

A companion to \texttt{tire\_fiber\_data.tex}: that note enumerated
fiber distributions in the spoke-only setting (no chords in the inner
outerplanar graph $O$).  This note investigates what happens when $O$
has chords --- specifically when the surrounding $G$ does \emph{not}
introduce Steiner vertices inside $B_{\mathrm{in}}$, so that each face
of $O$ in the tire's embedding remains a single face of $G$.
Script: \texttt{experiments/tire\_fiber\_chords.py}.  Output:
\texttt{experiments/tire\_fiber\_chords\_data.txt}.

\section*{The modelling choice that chords force on us}

In the spoke-only enumeration (\texttt{tire\_fiber\_data.tex}) the
inner-spoke part of $T'_{f'}$ was simply $k$ pendant edges, one per
$B_{\mathrm{in}}$ edge.  That picture corresponds to a surrounding
maximal planar graph $G$ that \emph{aggressively triangulates} the
disk inside $B_{\mathrm{in}}$ using interior (Steiner) vertices not in
$V(T)$, so each $B_{\mathrm{in}}$ edge gets its own dedicated inside-$O$
sub-triangle.

When we add chords to $O$, there is a second axis of choice: does $G$
sub-triangulate the resulting $O$-faces, or not?  Two extreme models:

\begin{itemize}
  \item \textbf{Steiner-rich (SR).}  $G$ adds whatever vertices and
        edges it needs inside each $O$-face so that every
        $B_{\mathrm{in}}$ edge has its own sub-triangle.  The
        inner-spoke graph is $k$ pendants \emph{regardless of the
        chord set} --- chords are invisible to $T'_{f'}$.  This is the
        step-1 baseline.
  \item \textbf{Steiner-poor (SP).}  $G$ does not further
        sub-triangulate $O$-faces beyond what $O$ itself provides.
        Each face of $O$ becomes a single face of $G$, whose dual
        vertex has degree (in $G'$, and hence in $T'_{f'}$) equal to
        the number of $B_{\mathrm{in}}$ edges on the face boundary.
        Edge $3$-colourable only if \emph{every $O$-face has at most
        $3$ $B_{\mathrm{in}}$ edges}.
\end{itemize}

Real surrounding triangulations $G$ live somewhere between these two,
and the choice is not part of the tire structure itself --- it is a
property of how $G$ embeds the tire.  But the two extremes already
say something useful about chord effects.

\section*{Side-by-side data}

\begin{center}
\scriptsize
\begin{tabular}{l r | r r r | r r r}
\toprule
& & \multicolumn{3}{c|}{\textbf{Steiner-rich}}
  & \multicolumn{3}{c}{\textbf{Steiner-poor}}\\
case & $n$ & $P_e$ & $|\pi_D(\mathcal{C})|/3^k$ & $|\pi_U(\mathcal{C})|/3^m$
            & $P_e$ & $|\pi_D(\mathcal{C})|/3^k$ & $|\pi_U(\mathcal{C})|/3^m$ \\
\midrule
(4,4) no chord                & 8  & 258  & 81/81  & 81/81  & 0    & 0/81   & 0/81  \\
(4,4) chord (0,2), faces 2+2  & 8  & 258  & 81/81  & 81/81  & 144  & 36/81  & 54/81 \\
(5,5) no chord                & 10 & 1026 & 243/243 & 243/243 & 0   & 0/243  & 0/243 \\
(5,5) chord (0,2), faces 2+3  & 10 & 1026 & 243/243 & 243/243 & 240 & 36/243 & 90/243 \\
(5,5) chord (0,3), faces 3+2  & 10 & 1026 & 243/243 & 243/243 & 240 & 36/243 & 90/243 \\
(6,6) no chord                & 12 & 4098 & 729/729 & 729/729 & 0   & 0/729  & 0/729 \\
(6,6) chord (0,3) antipodal, 3+3 & 12 & 4098 & 729/729 & 729/729 & 408  & 36/729  & 90/729  \\
(6,6) chords (0,2)(3,5), 2+2+2 & 12 & 4098 & 729/729 & 729/729 & 1536 & 216/729 & 456/729 \\
(3,4) no chord                & 7  & 126  & 78/81  & 27/27  & 0    & 0/81   & 0/27 \\
(3,4) chord (0,2)             & 7  & 126  & 78/81  & 27/27  & 48   & 36/81  & 24/27 \\
(6,4) no chord                & 10 & 1026 & 81/81  & 549/729 & 0   & 0/81   & 0/729 \\
(6,4) chord (0,2)             & 10 & 1026 & 81/81  & 549/729 & 480 & 36/81  & 342/729 \\
\bottomrule
\end{tabular}
\end{center}

\noindent
($\pi_D$ projects $\sigma$ onto the $k$ inner-direction spoke positions
on the dual cycle; $\pi_U$ onto the $m$ outer-direction positions.
``Chord (a,b)'' means $O = B_{\mathrm{in}} \cup \{v_a v_b\}$.)

\section*{Observations}

\begin{obs}[SP is empty without chords for $k > 3$]
\label{obs:sp-empty}
If $O$ has no chord and $k \geq 4$, then $O$ has a single inner face
with $k$ incident $B_{\mathrm{in}}$ edges.  Under SP this is a single
$G'$-vertex of degree $k$, and proper edge $3$-colouring needs $k$
distinct colours at that vertex, which is impossible for $k \geq 4$.
\textbf{So the SP model forces chords once $k \geq 4$}: tires whose
inner boundary is longer than $3$ are only feasible if $O$ already
contains enough chords to break every face down to $\leq 3$
$B_{\mathrm{in}}$ edges.
\end{obs}

\noindent
This is the structural reason chord-aware modelling matters.  Under
SR, you can pretend chord sets do not exist; under SP, you cannot.

\begin{obs}[Adding chords \emph{enlarges} the SP support]
\label{obs:more-chords-more-support}
Comparing the two $(6,6)$ chord rows:
\begin{itemize}
  \item Antipodal $(0,3)$ chord: face sizes $3{+}3$, gives
        $P_e = 408$, $\pi_D$ support $36/729 \approx 4.9\%$.
  \item Two chords $(0,2),(3,5)$: face sizes $2{+}2{+}2$, gives
        $P_e = 1536$, $\pi_D$ support $216/729 \approx 29.6\%$.
\end{itemize}
Smaller faces $=$ weaker constraints $=$ more realisable
configurations.  This is the opposite of what one might expect at
first glance (``more chords $=$ more edges $=$ more constraints'') ---
each chord splits a hard constraint into easier ones.
\end{obs}

\begin{obs}[$\pi_D$ support depends only on chord structure, not on $m$]
\label{obs:pi-D-m-independent}
For chord set $\{(0,2)\}$ in $k = 4$, the $\pi_D$ support is
$36/81 = 4/9$ for every tested $m \in \{3, 4, 6\}$.  This is structural:
the chord constraint pins down which spoke-quadruples are realisable
without involving the outer cycle at all.  The $\pi_U$ support, by
contrast, varies with $m$.
\end{obs}

\begin{obs}[SP support shrinks the $\pi_U$ support too]
\label{obs:u-shrinks}
Even though chord constraints live on the inner side, they propagate
to the outer side via the cycle.  For $(m, k) = (4, 4)$:
\begin{itemize}
  \item SR: $\pi_U$ saturates $3^4$.
  \item SP with chord $(0,2)$: $\pi_U = 54/81 = 2/3$.
\end{itemize}
The chord-induced constraints on D-position spokes propagate around
the cycle through the proper-edge-colouring constraint, reducing the
set of cycle colourings, hence reducing $\pi_U$ as well.
\end{obs}

\section*{Implications for the chain-pigeonhole step}

In the spoke-only / SR step-1 analysis we observed that whenever
$|B_{\mathrm{out}}| \geq |B_{\mathrm{in}}|$ the inner-side projection
saturates the full ring-coloring universe $3^k$, so chain
compatibility on a shared cycle was trivially nonempty whenever at
least one of the two adjacent tires had the longer non-shared
boundary.

Under SP this clean saturation breaks down on the chord side:

\begin{itemize}
  \item For a chorded tire, $\pi_D$ never reaches $3^k$ --- the
        chord constraints permanently rule out some inner-spoke
        configurations.  The realisable set is small but non-empty
        (e.g., $36/81$ at $k = 4$).
  \item The shrinkage is the same regardless of $m$ on the inner
        (D) side, so making the outer boundary very long does
        \emph{not} help recover saturation on the chord side.
  \item Two adjacent SP tires sharing a cycle $\gamma$ both have
        shrunk projections; whether they intersect is now a real
        question.  E.g., projecting $\pi_D$ down to a $4/9$-fraction
        of $\{1,2,3\}^k$ from each side: do the two $4/9$-sets always
        meet?
\end{itemize}

\noindent
That last question is the natural follow-up experiment (and is
step~2 of the action items).

\subsection*{The structural fact worth stating}

\begin{obs}[Steiner-poor chord-rule]
\label{obs:sp-chord-rule}
Under the SP model, a tire $T$ is edge-$3$-colourable on its face
connector $T'_{f'}$ iff its inner outerplanar graph $O$ \emph{already
triangulates the inside of $B_{\mathrm{in}}$ up to faces of $B_{\mathrm{in}}$-
size $\leq 3$}.  Equivalently: $O$ must contain enough chords that
every $O$-face has at most $3$ $B_{\mathrm{in}}$ edges on its boundary.
\end{obs}

\noindent
For a level-source BFS decomposition with $O$ chosen minimally (just
$B_{\mathrm{in}}$ itself, no chords), every tire with $k \geq 4$ fails
the SP edge-$3$-colouring test \emph{locally}.  Globally this is no
contradiction --- the surrounding maximal planar $G$ does add Steiner
vertices and edges inside $B_{\mathrm{in}}$, making us closer to SR
than to SP.  But the SP model is a worst-case envelope on what
purely local data implies, and shows that any structural argument that
ignores the inside-$B_{\mathrm{in}}$ triangulation is incomplete.

\section*{Caveats / what we did not compute}

\begin{enumerate}
  \item \textbf{Only the two extremes.}  Real $G$ triangulations lie
        between SR (every $B_{\mathrm{in}}$ edge gets its own
        Steiner-sub-triangle) and SP (no further sub-triangulation
        at all).  Intermediate cases (e.g., fan-triangulating one
        $O$-face from a $V(B_{\mathrm{in}})$ vertex without adding
        Steiner points) are not enumerated here, though the data
        suggests they sit between.
  \item \textbf{Outer side always SR.}  We have always treated the
        outer side ($B_{\mathrm{out}}$) as having no chord-induced
        structure --- $B_{\mathrm{out}}$ is just a cycle in the tire
        definition, so this is correct, but the analogous question
        ``what if the region outside $B_{\mathrm{out}}$ is poorly
        triangulated'' is not addressed.
  \item \textbf{$\Delta \leq 3$ on $O$ ignored.}  The chord
        configurations tested above include cases where chord
        endpoints have higher degree in $T$; the menagerie's
        $\Delta \leq 3$ constraint is not enforced.  This is fine
        for the SP model in isolation but matters when composing
        with the menagerie counting formulas.
  \item \textbf{Adjacency intersection (step 2).}  We have only
        computed single-tire fiber distributions.  The chain-
        pigeonhole step itself --- intersecting $\pi_D$ of one tire
        with $\pi_U$ of the adjacent tire on a shared cycle --- is
        deferred.
\end{enumerate}

\end{document}
