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\begin{document}

\title{Coloring Nested Tire Dual Graphs}

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}

\subjclass[2010]{Primary }

\keywords{plane graph, triangulation, plane depth, level edge, dual graph}

\date{}

\dedicatory{}

\begin{abstract}
This is a follow-up to \cite{bauerfeld-nested-tires}, which
establishes the basic vocabulary of tire graphs $T$ and dual
depth, along with the tire-component lemma and the tire-tread
partition theorem.  Building on those structural results, we
define the \emph{partial tire dual} $D(T)$ and analyse its
structure in the spoke-only case (a corona graph $C_{n+m} \circ
K_1$), give an edge-vertex coloring bijection that reduces
counting proper $3$-edge-colorings of $D(T)$ to counting proper
$3$-vertex-colorings of a cycle, and develop the
tire-annular-subgraph, face-connector, and inner/outer-spoke
structures in $G'$.  A concluding section records a
Latin-substructure conjecture for chain-pigeonhole compatibility
of adjacent tires.
\end{abstract}

\maketitle

\section{Introduction}

A classical theorem of Tait recasts the Four Colour Theorem in dual,
edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable
if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a
minimal counterexample to the Four Colour Theorem -- a smallest triangulation
admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
admitting no proper $3$-edge-colouring.

This paper is the second in a series studying that structure
through the lens of \emph{nested level duals}.  The foundational
vocabulary --- level sources, levels, the inner planar dual $G'$
and its dual depth, and tire graphs --- is developed in the
companion paper \cite{bauerfeld-nested-tires}; we refer to that
paper for those definitions and rely on them throughout.  In
particular we use, without restating, the notions of:
\begin{itemize}
  \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$;
  \item the inner planar dual $G'$
        (\cite[Definition~1.3]{bauerfeld-nested-tires});
  \item \emph{dual depth} $\delta_G(d_f)$
        (\cite[Definition~1.4]{bauerfeld-nested-tires});
  \item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$
        with outer/inner boundaries and annular edges
        (\cite[Definition~1.5]{bauerfeld-nested-tires});
  \item face/edge counts
        (\cite[Remark~1.6]{bauerfeld-nested-tires});
  \item the \emph{tire-component lemma}
        (\cite[Lemma~1.8]{bauerfeld-nested-tires}), which exhibits
        each connected component of $G'_d$ as a tire graph
        whose tire tread is the union of its depth-$d$ faces;
  \item the \emph{tire-tread partition theorem}
        (\cite[Theorem~1.9]{bauerfeld-nested-tires}), which shows
        the tire treads from a level source partition the bounded
        faces of $G$.
\end{itemize}

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.


\begin{definition}[Partial tire dual]
\label{def:partial-tire-dual}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
the sense of \cite[Definition~1.5]{bauerfeld-nested-tires}, and let
$F_{\mathrm{ann}}$
denote the set of triangular faces of $T$ in the tire tread
(the closed region between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$).  The \emph{partial
tire dual} of $T$, written $D(T)$, is the graph defined as follows.

\emph{Vertices.}
\begin{enumerate}
  \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
        \emph{interior vertex} $d_f$ of $D(T)$.
  \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
        \emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
  \item[(V3)] For each occurrence of an edge in the closed walk
        $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
        a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$.  (When $O$ is
        $2$-connected each edge appears once; cut-vertices and
        bridges of $O$ may cause an edge or vertex to appear more
        than once.)
\end{enumerate}

\emph{Edges.}
\begin{enumerate}
  \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
        lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
        one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
        $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
        incident to $e$.
  \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
        $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
        $f \in F_{\mathrm{ann}}$ is the unique annular face incident
        to $e$.  The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
  \item[(E3)] For each occurrence of $e$ on the boundary walk
        $B_{\mathrm{in}}$, one edge
        $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
        $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
        on the side of that occurrence.  The leaf
        $\ell_e^{\mathrm{in}}$ has degree $1$.
\end{enumerate}
\end{definition}

\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png}
\caption{The partial tire dual $D(T)$ (purple squares + orange
diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$
and $k = 4$.  The ten interior vertices $d_f$ at the centroids of the
annular triangles form a single $10$-cycle (solid purple); each
boundary edge of the tire tread (either of $B_{\mathrm{out}}$ or
of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond)
attached to the unique annular face incident to it (dashed orange),
giving the structure $C_{10} \circ K_1$ of
Proposition~\ref{prop:partial-tire-dual-structure}.}
\label{fig:partial-tire-dual-example}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
$O$ has a bridge --- here a non-trivial edge cut connecting two
disjoint triangles.  $B_{\mathrm{out}}$ is a $4$-cycle on
$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
the bridge disconnects $O$).  Because both faces incident to the
bridge are annular triangles, the bridge contributes an
\emph{interior dual edge} (highlighted in red) rather than two
leaves; consequently the interior dual subgraph is no longer the
single $(n+m)$-cycle of
Proposition~\ref{prop:partial-tire-dual-structure}, but a theta
graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
annular faces) are joined by three internally vertex-disjoint paths
in $D(T)$.  Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
three for each triangle).}
\label{fig:partial-tire-dual-bridge}
\end{figure}

\begin{proposition}[Structure of $D(T)$ when the annular triangulation
is spoke-only]
\label{prop:partial-tire-dual-structure}
Suppose $B_{\mathrm{out}}$ is a simple cycle of length $n$, $O$ is a
$2$-connected outerplanar graph whose outer-face cycle
$B_{\mathrm{in}}$ has length $m$, and $E_{\mathrm{ann}}$ consists only
of \emph{spokes} (edges with one endpoint in $V(B_{\mathrm{out}})$ and
one in $V(B_{\mathrm{in}})$).  Then each face $f \in F_{\mathrm{ann}}$
has exactly one boundary edge (on $B_{\mathrm{out}}$ or
$B_{\mathrm{in}}$) and two interior annular edges, and consequently
$D(T)$ is isomorphic to the corona graph $C_{n+m} \circ K_1$: a cycle
of length $n + m$ on the interior vertices $\{d_f\}$, with one leaf
attached to each cycle vertex.

In particular, $|V(D(T))| = 2(n+m)$ and $|E(D(T))| = 2(n+m)$.
\end{proposition}

\begin{proof}
Each annular triangle $f$ in a spoke-only triangulation has the form
$\{x, y, z\}$ with $x \in V(B_{\mathrm{out}})$, $y \in V(B_{\mathrm{in}})$,
and $z$ also in $V(B_{\mathrm{out}}) \cup V(B_{\mathrm{in}})$.  Of its
three edges, the one between the two same-side vertices
($x$-$z$ if both on $B_{\mathrm{out}}$, or $y$-$z$ if both on
$B_{\mathrm{in}}$) is a boundary edge of the tire tread; the
other two edges are spokes.

So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes
shared with adjacent annular faces) and one leaf.  The induced subgraph
on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the
connectedness of the tire tread this forces it to be a single
cycle.  By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + m$,
and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof}


\begin{proposition}[Edge--vertex coloring bijection for $D(T)$]
\label{prop:edge-vertex-bijection}
Let $T$ be a tire graph satisfying the spoke-only hypothesis of
Proposition~\ref{prop:partial-tire-dual-structure} (so $D(T) \cong
C_{n+m} \circ K_1$).  Let $\Gamma \cong C_{n+m}$ be the interior
dual subgraph of $D(T)$ induced on the interior dual vertices
$\{d_f : f \in F_{\mathrm{ann}}\}$.  Then the number of proper
$3$-edge-colorings of $D(T)$ equals the number of proper
$3$-vertex-colorings of $\Gamma$, both given by
\[
    2^{n+m} + 2 \cdot (-1)^{n+m}.
\]
\end{proposition}

\begin{proof}
Write $L = n + m$, $\Gamma = C_L$.  We construct mutually inverse
bijections.

\emph{Step 1: proper $3$-edge-colorings of $D(T)$ $\leftrightarrow$
proper $3$-edge-colorings of $C_L$.}  Given a proper $3$-edge-coloring
$\chi$ of $D(T)$, the three edges incident to any $d_f$ carry three
distinct colors; in particular the two cycle edges incident to $d_f$
carry distinct colors, so $\chi|_{E(C_L)}$ is a proper $3$-edge-coloring
of $C_L$.  Conversely, given a proper $3$-edge-coloring $\psi$ of
$C_L$, the two cycle edges at any $d_f$ have distinct colors, so a
unique third color is available; assign that color to $d_f$'s leaf
edge.  The resulting extension to $D(T)$ is proper at every $d_f$ and
vacuously proper at every leaf (degree~$1$), and the two maps are
inverse to each other.  Therefore
\[
    \#\bigl\{\text{proper $3$-edge-colorings of } D(T)\bigr\}
    \;=\;
    \#\bigl\{\text{proper $3$-edge-colorings of } C_L\bigr\}.
\]

\emph{Step 2: proper $3$-edge-colorings of $C_L$ $\leftrightarrow$
proper $3$-vertex-colorings of $L(C_L) \cong C_L$.}  The line graph
$L(C_L)$ of a cycle of length $L$ is again a cycle of length $L$;
proper edge-colorings of $C_L$ are by definition proper vertex-colorings
of $L(C_L)$.

\emph{Step 3: count.}  The chromatic polynomial of the cycle is
$P(C_L, k) = (k-1)^L + (-1)^L (k-1)$; at $k = 3$ this gives
$2^L + 2 \cdot (-1)^L$.
\end{proof}

\begin{remark}
\label{rem:edge-vertex-corollary}
Proposition~\ref{prop:edge-vertex-bijection} reduces counting proper
$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings of
a single cycle, giving a closed form $2^{n+m} + 2(-1)^{n+m}$ that
depends only on $n+m$ (not on the specific spoke-only annular
triangulation, nor on the chord structure of $O$).  The count is
preserved under the corona-with-$K_1$ structure of
Proposition~\ref{prop:partial-tire-dual-structure} precisely because
each degree-$1$ leaf imposes no proper-edge-coloring constraint on
itself; its color is freely determined as the missing third color at
its attached interior vertex.
\end{remark}

\begin{definition}[Tire annular subgraph]
\label{def:tire-annular-subgraph}
Let $G$ be a maximal planar graph with embedding $\Pi_G$ and inner
planar dual $G'$ (as in \cite[Definition~1.3]{bauerfeld-nested-tires} above).  Let
$T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire
graph (\cite[Definition~1.5]{bauerfeld-nested-tires}), and let
$F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces.
The \emph{tire annular subgraph} of $T$ in $G'$ is
\[
  T'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr],
\]
the subgraph of $G'$ induced on the dual vertices corresponding to the
annular faces of $T$.  We equip $T'_{\mathrm{ann}}$ with the planar
embedding inherited from $G'$ (which, by deletion of vertices outside
the annulus, remains a planar embedding of $T'_{\mathrm{ann}}$ in the
sense of $\Pi_G$).
\end{definition}

\begin{definition}[Tire annular face connector]
\label{def:tire-annular-face-connector}
With $G, G', T$ as in
Definition~\ref{def:tire-annular-subgraph}, let $f'$ be a face of the
tire annular subgraph $T'_{\mathrm{ann}}$ in its inherited embedding,
and let $V(f') \subseteq V(T'_{\mathrm{ann}})$ denote the set of
vertices on the boundary walk of $f'$.  The \emph{tire annular face
connector at $f'$} is the subgraph
\[
  T'_{f'} \;:=\; \bigl(\,V(f') \cup N_{G'}(V(f'))\,,\;
                       \{\,e \in E(G') : e \text{ is incident to } V(f')\,\}\,\bigr)
                \;\subseteq\; G',
\]
i.e.\ the subgraph of $G'$ on the closed $G'$-neighborhood of $V(f')$
together with every $G'$-edge incident to $V(f')$.
\end{definition}

\begin{definition}[Inner and outer spokes]
\label{def:spokes}
With $T'_{f'}$ as in
Definition~\ref{def:tire-annular-face-connector}, regard $f'$ as an
open region of $|\Pi_G|$ and write $\overline{f'}$ for its closure.
The vertices of $V(T'_{f'}) \setminus V(f')$ lie in $|\Pi_G| \setminus
\overline{f'}$ or in $f'$ (never on $\partial f'$, since the boundary
walk of $f'$ is by definition the set $V(f')$).  Partition
\[
  V(T'_{f'}) \setminus V(f') \;=\;
  V_{\mathrm{out}}(T'_{f'}) \,\sqcup\, V_{\mathrm{in}}(T'_{f'})
\]
where
\begin{align*}
  V_{\mathrm{out}}(T'_{f'}) &:= \{\, v \in V(T'_{f'}) \setminus V(f')
                              \;:\; v \notin \overline{f'} \,\}, \\
  V_{\mathrm{in}}(T'_{f'})  &:= \{\, v \in V(T'_{f'}) \setminus V(f')
                              \;:\; v \in f' \,\}.
\end{align*}
The elements of $V_{\mathrm{out}}(T'_{f'})$ are the \emph{outer
spokes} of $T'_{f'}$ (vertices not in $V(f')$ that lie outside the
region bounded by $f'$); the elements of $V_{\mathrm{in}}(T'_{f'})$
are the \emph{inner spokes} of $T'_{f'}$ (vertices not in $V(f')$
that lie inside the region bounded by $f'$).
\end{definition}

\begin{remark}
\label{rem:facial-dual-spoke-only}
In the spoke-only setting of
Proposition~\ref{prop:partial-tire-dual-structure}, the tire annular
subgraph is $T'_{\mathrm{ann}} = \Gamma \cong C_{n+m}$
(Proposition~\ref{prop:edge-vertex-bijection}).  This cycle has exactly
two faces in its inherited embedding -- one on each side of the cycle
in $\Pi_G$ -- and both face boundaries traverse all $n+m$ vertices, so
$V(f') = V(\Gamma)$ for either choice of $f'$.  Each interior dual
vertex $d_f$ has $G'$-degree $3$ (since $G$ is a triangulation), of
which two edges lie in $\Gamma$ (cycle edges) and one edge points to a
single non-annular face of $G$.  Consequently $T'_{f'}$ has $n + m$
interior vertices plus the non-annular face vertices to which they
connect, and is independent of the choice of $f'$.  When $G$ consists
only of the tire $T$ together with one source-side face inside
$B_{\mathrm{out}}$ and one $O$-side face inside $B_{\mathrm{in}}$,
$T'_{f'}$ recovers the planar dual of $T$ itself.
\end{remark}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{notes/fig_facial_dual_choices.png}
\caption{\label{fig:facial-dual-choices}
The bridge case: $T'_{\mathrm{ann}} = \theta(1, 3, 3)$ has three faces
$A, B, C$ in its inherited embedding, with respective vertex sets
$V(A) = \{v_0, \dots, v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and
$V(C) = \{v_0, v_3, v_4, v_5\}$.  In the surrounding maximal planar
$G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing
the bridge edge) have all three $G'$-edges inside $T'_{\mathrm{ann}}$,
while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes
one $G'$-edge to an external non-annular neighbor $u_i$.  Each panel
highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$,
gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm{ann}}$,
and red squares are external $G'$-neighbors $u_i$.  The choice of
face $f'$ controls which external neighbors $u_i$ are pulled into
$T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$
and face $C$ pulls in $u_4, u_5$).}
\end{figure}

\section{A conjectural Latin-style substructure}
\label{sec:latin-conjecture}

Empirical enumeration (notes \texttt{tire\_fiber\_data.tex},
\texttt{tire\_fiber\_chords.tex}, \texttt{tire\_fiber\_step2.tex},
\texttt{tire\_fiber\_step2\_large.tex}) of edge $3$-coloring
distributions on the tire annular face connector $T'_{f'}$ across
$46$ adjacent-tire pairs at $|\gamma| \in \{3, 4, 5, 6, 9, 12\}$
suggests that the chain-pigeonhole step on a shared cycle always
succeeds.  The data points to a structural mechanism: every
edge-$3$-colourable tire admits at least one ``Latin-flavoured''
boundary configuration, and adjacent tires share this same
substructure on their common cycle.

Concretely, fix a tire $T$ with inner outerplanar graph $O$ on $V(B_{\mathrm{in}})$
and let $F(O)$ be the set of $O$-faces (in the tire's plane embedding,
not counting the outer face $B_{\mathrm{in}}$).  For each $O$-face
$f \in F(O)$, let $E_{\mathrm{in}}(f) \subseteq E(B_{\mathrm{in}})$
denote the set of $B_{\mathrm{in}}$ edges on $f$'s boundary.  In the
Steiner-poor surrounding triangulation (where each $O$-face is a
single face of $G$ and dualises to a single $G'$-vertex of degree
$|E_{\mathrm{in}}(f)|$ in $T'_{f'}$), proper edge $3$-colouring of
$T'_{f'}$ requires every $O$-face to have $|E_{\mathrm{in}}(f)| \leq 3$.

Let $\sigma_{B_{\mathrm{in}}}$ denote the spoke colouring restricted to
the $|V(B_{\mathrm{in}})|$ inner-direction spoke positions on the dual
annular cycle (equivalently: $\sigma$ indexed by $E(B_{\mathrm{in}})$).
Define the \emph{Latin-flavoured set} on $\gamma = B_{\mathrm{in}}$ as
\[
   \mathcal{L}(B_{\mathrm{in}}, O)
   \;:=\; \bigl\{\,\sigma : E(B_{\mathrm{in}}) \to \{1,2,3\}
     \;\big|\; \sigma\bigl|_{E_{\mathrm{in}}(f)}\text{ is a permutation
     of }\{1,2,3\}\text{ for every } f \in F(O)\,\bigr\}.
\]
That is, on every $O$-face's $B_{\mathrm{in}}$-edge boundary, all three
colours appear exactly once (forcing $|E_{\mathrm{in}}(f)| = 3$ for
each face --- the maximally constrained case).

\begin{conjecture}[Latin-substructure conjecture]
\label{conj:latin}
For any Steiner-poor edge-$3$-colourable tire $T$ with inner
outerplanar graph $O$ such that every $O$-face has exactly $3$
$B_{\mathrm{in}}$-edges, the realisable inner-spoke projection
$\pi_D(\mathcal{C}(T'_{f'}))$ contains $\mathcal{L}(B_{\mathrm{in}}, O)$
as a subset.  Moreover, $\mathcal{L}(B_{\mathrm{in}}, O)$ is invariant
under the $S_3$ action on colours and has size at least $3! = 6$.
\end{conjecture}

\begin{conjecture}[Chain-pigeonhole compatibility from Latin substructure]
\label{conj:chain-latin}
Adjacent tires $T_1, T_2$ sharing a cycle $\gamma$ admit a joint
edge $3$-colouring whenever their respective inner-outerplanar
structures $O^{(1)}, O^{(2)}$ both satisfy
Conjecture~\ref{conj:latin}.  Equivalently:
$\pi_D^{(1)}(\mathcal{C}(T_1)) \cap \pi_U^{(2)}(\mathcal{C}(T_2))
\supseteq \mathcal{L}(\gamma, O^{(1)}) \cap \mathcal{L}(\gamma,
O^{(2)})$, and this last intersection is non-empty whenever the two
face partitions of $E(\gamma)$ induced by $O^{(1)}, O^{(2)}$ share a
common ``Latin completion.''
\end{conjecture}

The structural origin of these conjectures is the empirical
observation that the smallest tested intersections on $\gamma$ are
always exactly the $3!$ permutations of a single canonical pattern
in which each $O$-face on $\gamma$'s side receives a permutation of
$\{1,2,3\}$.  For example, at $|\gamma| = 12$ with $O^{(1)}$ given by
the chord matching $\{(0,3),(4,7),(8,11)\}$ (face structure $\{0,1,2\}
\sqcup \{4,5,6\} \sqcup \{8,9,10\} \sqcup \{3,7,11\}$), the canonical
pattern $(1,2,3,2,2,1,3,3,2,3,1,1)$ assigns the permutation $(1,2,3)$
to the first face, $(2,1,3)$ to the second, $(2,3,1)$ to the third,
and $(2,3,1)$ to the fourth.  Every face receives all three colours.

A proof of Conjecture~\ref{conj:latin} would convert the
chain-pigeonhole compatibility step into a structural theorem on
$T'_{f'}$: it is not the rough abundance of valid spoke configurations
that lets adjacent tires meet, but a specific Latin-square-flavoured
substructure dictated by the face partition of each tire's inner
outerplanar graph.  See \texttt{notes/tire\_fiber\_step2\_large.tex}
for the data underlying this conjecture and
\texttt{experiments/tire\_fiber\_counterexample\_search.log} for the
ongoing automated search.

\begin{thebibliography}{9}

\bibitem{bauerfeld-depth}
E.~Bauerfeld,
\emph{Plane Depth},
manuscript (math-research repository), 2026.

\bibitem{bauerfeld-nested-tires}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.

\end{thebibliography}

\end{document}
