Add medial tire decomposition paper

papers/nested_tire_decompositions_of_plane_triangulations/paper.tex

@@ -52,9 +52,8 @@ $G$ induces a BFS layering of $G$ and endows the inner planar dual
 $G'$ with a \emph{dual depth} grading.  The basic object of study is
 the \emph{tire graph} $T$ --- a plane graph whose outer and inner
 boundaries bound a closed planar region, the \emph{tire tread} $R$,
-triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$.  We define
-medial tire graphs and prove a basic colour-count bound for their
-annular medial cycle.  Our main structural results are the
+triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$.  Our main
+structural results are the
 \emph{tire-component lemma}, the \emph{tire-tread partition theorem},
 and the rooted \emph{tire-tree decomposition}, which together organize
 the bounded faces of $G$ into nested tire treads.
@@ -283,52 +282,6 @@ corresponding retained tire edge.}
 \label{fig:medial-tire-example}
 \end{figure}
 
-\begin{theorem}[Annular medial colour bound]
-\label{thm:annular-medial-colour-bound}
-Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph with
-non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
-Let $A(T)$ be the subgraph of $M_{\mathrm{tire}}(T)$ induced by the
-annular medial vertices.  For a graph $H$, write
-$\operatorname{Col}_3(H)$ for the set of proper $3$-vertex-colourings
-of $H$.  Then $A(T)$ is a cycle and
-\[
-    |\operatorname{Col}_3(M_{\mathrm{tire}}(T))|
-    \;\leq\; |\operatorname{Col}_3(A(T))|.
-\]
-\end{theorem}
-
-\begin{proof}
-Since the tread is a triangulated annulus with no vertices in its
-interior, each annular face has exactly one boundary edge, lying either
-on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
-edges.  As the annular faces are traversed cyclically around the tread,
-consecutive faces share one annular edge.  Equivalently, the annular
-edges occur in a cyclic order in which each annular face contains two
-consecutive annular edges.  Hence the subgraph of
-$M_{\mathrm{tire}}(T)$ induced by the annular medial vertices is a
-cycle.
-
-Consider the restriction map from proper $3$-colourings of
-$M_{\mathrm{tire}}(T)$ to colourings of this annular medial cycle
-$A(T)$.  We claim that this map is injective.  Let $x$ be a
-non-annular medial vertex.  Then $x$ corresponds to an edge of
-$B_{\mathrm{out}}$ or $B_{\mathrm{in}}$, since the chords of $O$ were
-omitted before forming $M_{\mathrm{tire}}(T)$.  This boundary edge is
-incident to a unique annular face of $T^{\circ}$, and the other two
-edges of that face are annular edges.  Therefore $x$ is adjacent in
-$M_{\mathrm{tire}}(T)$ to the two annular medial vertices corresponding
-to those two annular edges.
-
-Those two annular medial vertices are adjacent to each other, because
-their annular edges are consecutive on the same triangular annular
-face.  In any proper $3$-colouring they therefore receive two distinct
-colours, and $x$ is forced to receive the remaining third colour.
-Thus every non-annular medial vertex has its colour uniquely determined
-by the colouring of $A(T)$.  Two colourings of $M_{\mathrm{tire}}(T)$
-with the same restriction to $A(T)$ are identical, so the restriction
-map is injective.  The stated inequality follows.
-\end{proof}
-
 \begin{remark}
 \label{rem:tire-counts}
 Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$.  By