diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 28e3cfa..23c4271 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -17,6 +17,8 @@ \newlabel{rem:tire-component-degenerate}{{1.10}{6}} \newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}} \newlabel{thm:inner-dual-outerplanar}{{1.12}{7}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\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.}}{8}{}\protected@file@percent } +\newlabel{fig:inner-dual-disk-case}{{3}{8}} \citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals} \bibcite{bauerfeld-depth}{1} @@ -26,7 +28,9 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{8}} -\newlabel{rem:bridge-case-theta}{{1.14}{8}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } -\gdef \@abspage@last{8} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with a single ``bridge''-style chord. Outer boundary $B_{\mathrm {out}}$ and inner boundary $B_{\mathrm {in}}$ are concentric hexagons (red). The annular region is triangulated by spokes (grey) and one extra interior annular edge between two inner vertices (dashed grey). The inner dual $\Gamma $ (blue) consists of $12$ dual vertices at the $12$ annular face centroids, connected as a Hamilton cycle around the annulus, plus one chord (dashed blue) corresponding to the extra interior edge. 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The dual graph of such a polygon -triangulation is a tree (a classical fact: a triangulation of a -$p$-gon with no interior vertex has $p - 2$ triangles and $p - 3$ -diagonals, and the diagonals' adjacency graph is a tree). Trees -are outerplanar. +Let $v_0$ be the degenerate-boundary vertex (the apex) and let +$k = |B_{\mathrm{non-deg}}|$ be the length of the non-degenerate +boundary cycle. The triangulation of $R$ is a \emph{fan} of $k$ +triangles around $v_0$: each triangle has the form $\{v_0, u_i, +u_{i+1}\}$ where $u_1, \dots, u_k$ are the boundary-cycle vertices +in cyclic order. Each triangle has two spoke edges (= the two +edges incident to $v_0$, shared with the two neighbouring fan +triangles) and one boundary edge (in $B_{\mathrm{non-deg}}$, +contributing a leaf in $D(T)$ but no edge in $\Gamma$). Hence +every $d_f$ has $\Gamma$-degree exactly $2$, and $\Gamma$ is a +single cycle of length $k$. Cycles are outerplanar. + +See Figure~\ref{fig:inner-dual-disk-case} for the disk case +($k = 6$). + +\begin{figure}[h] +\centering +\begin{tikzpicture}[scale=1.4] + \def\R{1.8} + % apex + \node[circle, fill=black, inner sep=1.6pt, label={right:$v_0$}] (apex) at (0, 0) {}; + % boundary vertices (hexagon) + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \node[circle, fill=black, inner sep=1.3pt] (u\i) at (\ang:\R) {}; + } + % boundary cycle edges (non-degenerate boundary) + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[red, thick] (u\i) -- (u\j); + } + % spoke edges (annular interior) + \foreach \i in {0,...,5} { + \draw[gray] (apex) -- (u\i); + } + % dual vertices at triangle centroids + dual cycle + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\angmid}{60*\i + 90 + 30} + \pgfmathsetmacro{\rmid}{0.62*\R} + \node[circle, fill=blue!70!black, inner sep=1.6pt] (d\i) at (\angmid:\rmid) {}; + } + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[blue!70!black, very thick] (d\i) -- (d\j); + } + % Labels + \node[red] at (0, -\R - 0.3) {\small non-degenerate boundary $B_{\mathrm{non-deg}}$}; + \node[blue!70!black] at (\R + 1.0, 0.5) {\small dual cycle $\Gamma \cong C_6$}; + \node[blue!70!black] at (\R + 1.0, 0.2) {\small (outerplanar)}; + \node[gray] at (-\R - 0.4, 0.3) {\small spokes}; +\end{tikzpicture} +\caption{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.} +\label{fig:inner-dual-disk-case} +\end{figure} \medskip \emph{Case 2: $R$ is an annulus} (both $B_{\mathrm{out}}$ and @@ -580,6 +635,71 @@ on the outer face (the unbounded region outside the circle), making $\Gamma$ outerplanar. $\square$ \end{proof} +\begin{figure}[h] +\centering +\begin{tikzpicture}[scale=1.35] + \def\Rout{2.0} + \def\Rin{1.05} + % Boundary cycles + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \node[circle, fill=black, inner sep=1.2pt] (uo\i) at (\ang:\Rout) {}; + } + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ang}{60*\i + 90 + 30} + \node[circle, fill=black, inner sep=1.2pt] (ui\i) at (\ang:\Rin) {}; + } + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[red, thick] (uo\i) -- (uo\j); + \draw[red!60!white, thick] (ui\i) -- (ui\j); + } + % Annular edges: spokes (each outer vertex connects to 2 inner) + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i,6)} + \pgfmathtruncatemacro{\k}{mod(\i+5,6)} + \draw[gray] (uo\i) -- (ui\j); + \draw[gray] (uo\i) -- (ui\k); + } + % Highlight one bridge-like annular interior edge — between two inner vertices + % (For illustration we use the "bridge" between inner i=0 and i=3) + \draw[gray, dashed, thick] (ui0) to[bend right=15] (ui3); + % Dual: 12 annular triangles → 12 dual vertices arranged between + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ango}{60*\i + 90 - 15} + \pgfmathsetmacro{\rmido}{0.5*\Rout + 0.5*\Rin} + \node[circle, fill=blue!70!black, inner sep=1.4pt] (do\i) at (\ango:\rmido) {}; + \pgfmathsetmacro{\angi}{60*\i + 90 + 15} + \node[circle, fill=blue!70!black, inner sep=1.4pt] (di\i) at (\angi:\rmido) {}; + } + % Dual cycle: do0 - di0 - do1 - di1 - ... around + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[blue!70!black, very thick] (do\i) -- (di\i); + \draw[blue!70!black, very thick] (di\i) -- (do\j); + } + % Chord for the bridge (one chord across the dual cycle) + \draw[blue!70!black, very thick, dashed] (di0) to[bend left=20] (di3); + % Labels + \node[red] at (0, \Rout + 0.35) {\small $B_{\mathrm{out}}$}; + \node[red!60!white] at (0, -\Rin + 0.15) {\small $B_{\mathrm{in}}$}; + \node[blue!70!black] at (\Rout + 0.85, 0.55) {\small Hamilton walk}; + \node[blue!70!black] at (\Rout + 0.85, 0.25) {\small + non-crossing}; + \node[blue!70!black] at (\Rout + 0.85, -0.05) {\small chord}; +\end{tikzpicture} +\caption{Case 2 ($R$ = annulus) with a single ``bridge''-style +chord. Outer boundary $B_{\mathrm{out}}$ and inner boundary +$B_{\mathrm{in}}$ are concentric hexagons (red). The annular +region is triangulated by spokes (grey) and one extra interior +annular edge between two inner vertices (dashed grey). The +inner dual $\Gamma$ (blue) consists of $12$ dual vertices at the +$12$ annular face centroids, connected as a Hamilton cycle around +the annulus, plus one chord (dashed blue) corresponding to the +extra interior edge. All $12$ vertices lie on the outer face of +the chord-augmented cycle, so $\Gamma$ is outerplanar.} +\label{fig:inner-dual-annulus-case} +\end{figure} + \begin{remark} \label{rem:hamilton-cycle-spoke-only} In the \emph{spoke-only} case (Definition~\ref{def:tire-graph} with