Add Medial Tire Cuts paper with walk-depth labelling and cut

New paper "Medial Tire Cuts" citing the medial tire decompositions
paper. States the goal of decomposing the medial graph into a tree of
3-faces, and gives the walk-depth labelling-and-cut procedure for a
single full medial tire graph: a cut duplicates the annular vertex
where a face's tooth traversal closes (planar unzip), reducing the
inner faces to teeth.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_cuts/paper.tex

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+%% filename: amsart-template.tex
+%% American Mathematical Society
+%% AMS-LaTeX v.2 template for use with amsart
+%% ====================================================================
+
+\documentclass{amsart}
+
+\usepackage{amssymb}
+\usepackage{graphicx}
+\usepackage{tikz}
+\usetikzlibrary{backgrounds}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{conjecture}[theorem]{Conjecture}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{xca}[theorem]{Exercise}
+
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+
+\numberwithin{equation}{section}
+
+\begin{document}
+
+\title{Medial Tire Cuts}
+
+%    author one information
+\author{Eric Bauerfeld}
+\address{}
+\curraddr{}
+\email{}
+\thanks{}
+
+\subjclass[2010]{Primary }
+
+\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}
+
+\date{}
+
+\dedicatory{}
+
+\begin{abstract}
+Starting from the medial tire decomposition of a plane triangulation, we
+study the cuts that medial tires make in the full medial graph.  We will
+show how to use medial tires to decompose the medial graph into a tree of
+three faces.
+\end{abstract}
+
+\maketitle
+
+\section{Introduction}
+
+This paper builds on the medial tire decomposition
+of~\cite{bauerfeld-medial-tire}.  For a plane triangulation $G$ with
+fixed embedding we use freely the terminology and notation introduced
+there: the full medial graph $M(G)$, its decomposition into full medial
+tire graphs $\mathsf{M}(T)$ indexed by the treads $T$ of the tire tree
+$\mathcal{T}(G,S)$ at a level source $S$, the annular medial cycle
+$A(T)$, and the boundary medial vertex sets.
+
+We will show how to use medial tires to decompose the medial graph into
+a tree of three faces.
+
+\section{Cutting a full medial tire graph}
+
+We first describe a procedure that simultaneously \emph{labels} and
+\emph{cuts} a single full medial tire graph $\mathsf{M}(T)$ so that,
+after the cuts, the only faces are the outer face and $3$-faces
+(triangles)---the teeth of~\cite{bauerfeld-medial-tire}.  The labelling
+assigns to each tooth an integer \emph{walk depth}; the cuts break the
+cyclic adjacencies of the teeth so that what remains is a tree of
+$3$-faces.
+
+By a \emph{cut} we mean the duplication of a single vertex of
+$\mathsf{M}(T)$: the vertex is split into two copies and the embedding is
+slit open along it (a planar unzip), separating the faces that meet only
+at that vertex.  A cut therefore reduces the number of bounded faces that
+are not teeth.
+
+Throughout we use the teeth, up and down teeth, apexes, bites, the
+annular medial cycle $A(T)$, and the auxiliary plane graph $B(T)$
+of~\cite{bauerfeld-medial-tire}.  Each tooth is a $3$-face of
+$\mathsf{M}(T)$, and the inner faces of $B(T)$ (the root face and the
+bite inner-gap faces) are the larger faces to be cut into teeth.
+
+\begin{definition}[Walk-depth labelling and cut]
+\label{def:walk-depth-cut}
+Let $\mathsf{M}(T)$ be a full medial tire graph.  Assign walk depths and
+cuts as follows.
+\begin{enumerate}
+  \item Pick an arbitrary up tooth, the \emph{entry tooth}.  It has walk
+  depth $d$.
+  \item Traverse all the teeth that bound the inner face incident to the
+  entry tooth clockwise until we reach the entry tooth, incrementing the
+  walk depth by $1$ for each tooth traversed.  (The \emph{inner face
+  incident to the entry tooth} is the inner face of $B(T)$ whose boundary
+  contains the annular edge of $A(T)$ carrying the entry tooth.)
+  \item When you reach the last tooth in the face, perform a \emph{cut}
+  by duplicating the annular vertex at which the traversal closes---the
+  annular vertex of $A(T)$ shared by the last tooth and the entry tooth.
+  \item Find the tooth $t$ with the highest walk depth which is a member
+  of a bite.
+  \item If $t$ is incident to a face $F$ with unlabelled teeth, traverse
+  the teeth in $F$ starting from $t$ in the direction of the tooth
+  incident to $t$ which is unlabelled, and increment the walk depth by
+  $1$ as you travel.  (Here a tooth is \emph{incident to $t$} when it
+  shares an annular vertex of $A(T)$ with $t$.)
+  \item Repeat steps (3)--(5) until all teeth have been labelled.
+\end{enumerate}
+\end{definition}
+
+\begin{thebibliography}{9}
+
+\bibitem{bauerfeld-medial-tire}
+E.~Bauerfeld,
+\emph{Medial Tire Decompositions of Plane Triangulations},
+manuscript (math-research repository), 2026.
+
+\end{thebibliography}
+
+\end{document}