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Add medial tire decomposition paper

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papers/nested_tire_decompositions_of_plane_triangulations/paper.aux
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\citation{dvorak-lidicky-cones}
\citation{heesch-untersuchungen}
\citation{robertson-sanders-seymour-thomas}
\citation{robertson-sanders-seymour-thomas}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent }
\citation{robertson-sanders-seymour-thomas}
\newlabel{def:dual}{{1.3}{2}}
\newlabel{def:dual-depth}{{1.4}{2}}
\newlabel{def:dual-component}{{1.5}{2}}
\newlabel{def:tire-graph}{{1.6}{2}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent }
\newlabel{fig:dual-depth}{{1}{3}}
\newlabel{def:tire-graph}{{1.6}{3}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent }
\newlabel{fig:tire-example}{{2}{4}}
\newlabel{def:medial-tire-graph}{{1.7}{4}}
\newlabel{thm:annular-medial-colour-bound}{{1.8}{4}}
\newlabel{rem:tire-counts}{{1.8}{4}}
\newlabel{prop:no-level-d-pinch}{{1.9}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The medial tire graph for the tire in Figure\nonbreakingspace 2\hbox {}. The chord of $O$ is drawn faintly and omitted before taking the medial graph; medial edges between consecutive outer-boundary edges or consecutive inner-boundary edges are also omitted. Each medial vertex is placed at the midpoint of its corresponding retained tire edge.}}{5}{}\protected@file@percent }
\newlabel{fig:medial-tire-example}{{3}{5}}
\newlabel{rem:tire-counts}{{1.9}{5}}
\newlabel{prop:no-level-d-pinch}{{1.10}{6}}
\newlabel{lem:tire-component}{{1.11}{6}}
\citation{bauerfeld-depth}
\newlabel{lem:tire-component}{{1.10}{6}}
\citation{bauerfeld-depth}
\newlabel{thm:tread-partition}{{1.12}{8}}
\newlabel{rem:tire-component-degenerate}{{1.13}{8}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.14}{8}}
\newlabel{thm:inner-dual-outerplanar}{{1.15}{9}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{10}{}\protected@file@percent }
\newlabel{fig:inner-dual-disk-case}{{4}{10}}
\newlabel{thm:tread-partition}{{1.11}{7}}
\newlabel{rem:tire-component-degenerate}{{1.12}{8}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.13}{8}}
\newlabel{thm:inner-dual-outerplanar}{{1.14}{8}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{9}{}\protected@file@percent }
\newlabel{fig:inner-dual-disk-case}{{4}{9}}
\citation{bauerfeld-nested-tire-duals}
\citation{bauerfeld-nested-tire-duals}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.15}{10}}
\newlabel{rem:bridge-case-theta}{{1.16}{10}}
\newlabel{thm:tread-tree}{{1.17}{10}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong \Theta (1, 7, 7)$ is outerplanar.}}{11}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{5}{11}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}}
\newlabel{rem:bridge-case-theta}{{1.17}{11}}
\newlabel{thm:tread-tree}{{1.18}{12}}
\newlabel{rem:tree-multiple-children}{{1.19}{13}}
\newlabel{thm:tire-tree-decomposition}{{1.20}{13}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
\newlabel{fig:tire-tree-decomposition}{{6}{15}}
\newlabel{rem:tree-coloring-factorisation}{{1.21}{15}}
\newlabel{rem:tree-multiple-children}{{1.18}{12}}
\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
\bibcite{tait-original}{1}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3}
\bibcite{birkhoff-reducibility}{4}
\bibcite{birkhoff-lewis-chromatic}{5}
\bibcite{tutte-four-colour-conjecture}{6}
\newlabel{rem:tree-coloring-factorisation}{{1.20}{14}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
\bibcite{tutte-algebraic-colorings}{7}
\bibcite{tutte-chromatic-sums-1973}{8}
\bibcite{heesch-untersuchungen}{9}
@@ -62,5 +60,6 @@
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.19\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
\newlabel{fig:tire-tree-decomposition}{{6}{15}}
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papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk
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papers/nested_tire_decompositions_of_plane_triangulations/paper.tex
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@@ -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