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coloring_nested_tire_graphs: redraw Figure 4 properly

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Previous Figure 4 had two bugs:
  (1) Dual vertices were placed in arbitrary positions, not at
      annular triangle centroids.
  (2) The "bridge" chord didn't actually correspond to a bridge,
      since B_in was drawn as a single hexagonal cycle (which has
      no bridges). For a real bridge, O needs to be a barbell.

Redrawn as a clean spoke-only example:
  - B_out: hexagon (6 outer vertices u_0..u_5, red).
  - B_in: triangle (3 inner vertices w_0, w_1, w_2, light red).
  - V(O) = V(B_in), no chord of O, no bridge.
  - Triangulation: 9 spokes between outer and inner.
  - 9 annular triangles: 6 "outer-cap" + 3 "inner-cap".
  - Dual vertices placed using TikZ barycentric coordinates at
    each triangle's exact centroid.
  - Dual graph Γ ≅ C_9 (just a cycle, no chords for spoke-only).

The chord/bridge case isn't drawn directly in the figure but is
referenced via Remark 1.14, which already discusses the bridge
case (Θ(1,b,c) = Hamilton cycle + length-1 chord) textually.

This keeps the figure correct and unambiguous; readers wanting
the chord case can refer to the remark or the dual paper.

Page count: 9 → 10.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 02:02:56 -04:00
parent 6995a71d53
commit 44d0ed12b0
4 changed files with 101 additions and 87 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@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. All $12$ vertices lie on the outer face of the chord-augmented cycle, so $\Gamma $ is outerplanar.}}{9}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus, spoke-only). Outer boundary $B_{\mathrm {out}}$ a hexagon (red); inner boundary $B_{\mathrm {in}}$ a triangle (light red); $V(O) = V(B_{\mathrm {in}})$ with no chord of $O$, so the triangulation is built purely from spokes (grey) between outer and inner vertices. Nine annular triangles (six ``outer-cap'' triangles with one inner-vertex apex, three ``inner-cap'' triangles with one outer-vertex apex) tile the annulus. Each blue dot is the centroid of an annular triangle; adjacent dots are joined whenever the two corresponding triangles share a spoke. The resulting inner dual $\Gamma $ is the cycle $C_9$, manifestly outerplanar. For a tire graph with a bridge in $O$, an additional non-crossing chord appears in $\Gamma $ (see Remark\nonbreakingspace 1.14\hbox {}).}}{9}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{4}{9}} \newlabel{fig:inner-dual-annulus-case}{{4}{9}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
\newlabel{rem:bridge-case-theta}{{1.14}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
\gdef \@abspage@last{9} \gdef \@abspage@last{10}
papers/coloring_nested_tire_graphs/paper.log
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -637,66 +637,80 @@ making $\Gamma$ outerplanar. $\square$
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\begin{tikzpicture}[scale=1.35] \begin{tikzpicture}[scale=1.3]
\def\Rout{2.0} \def\Rout{2.0}
\def\Rin{1.05} \def\Rin{0.8}
% Boundary cycles % Outer hexagon vertices u_i at angles 90, 30, -30, -90, -150, 150
\foreach \i in {0,...,5} { \foreach \i in {0,...,5} {
\pgfmathsetmacro{\ang}{60*\i + 90} \pgfmathsetmacro{\ang}{90 - 60*\i}
\node[circle, fill=black, inner sep=1.2pt] (uo\i) at (\ang:\Rout) {}; \node[circle, fill=black, inner sep=1.3pt, label={\ang:\scriptsize $u_\i$}] (u\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) {};
} }
% Inner triangle vertices w_0 at 60, w_1 at -60, w_2 at 180
\node[circle, fill=black, inner sep=1.3pt, label={[label distance=-1pt]60:\scriptsize $w_0$}] (w0) at (60:\Rin) {};
\node[circle, fill=black, inner sep=1.3pt, label={[label distance=-1pt]-60:\scriptsize $w_1$}] (w1) at (-60:\Rin) {};
\node[circle, fill=black, inner sep=1.3pt, label={[label distance=-1pt]180:\scriptsize $w_2$}] (w2) at (180:\Rin) {};
% Outer boundary cycle (red)
\foreach \i in {0,...,5} { \foreach \i in {0,...,5} {
\pgfmathtruncatemacro{\j}{mod(\i+1,6)} \pgfmathtruncatemacro{\j}{mod(\i+1,6)}
\draw[red, thick] (uo\i) -- (uo\j); \draw[red, thick] (u\i) -- (u\j);
\draw[red!60!white, thick] (ui\i) -- (ui\j);
} }
% Annular edges: spokes (each outer vertex connects to 2 inner) % Inner boundary cycle (light red)
\foreach \i in {0,...,5} { \draw[red!55!white, thick] (w0) -- (w1) -- (w2) -- (w0);
\pgfmathtruncatemacro{\j}{mod(\i,6)} % Spokes (gray)
\pgfmathtruncatemacro{\k}{mod(\i+5,6)} \draw[gray] (u0) -- (w0);
\draw[gray] (uo\i) -- (ui\j); \draw[gray] (u1) -- (w0);
\draw[gray] (uo\i) -- (ui\k); \draw[gray] (u1) -- (w1);
\draw[gray] (u2) -- (w1);
\draw[gray] (u3) -- (w1);
\draw[gray] (u3) -- (w2);
\draw[gray] (u4) -- (w2);
\draw[gray] (u5) -- (w2);
\draw[gray] (u5) -- (w0);
% Dual vertices: 9 annular triangles, at centroids
% outer-caps (6): {u0,u1,w0}, {u1,u2,w1}, {u2,u3,w1}, {u3,u4,w2}, {u4,u5,w2}, {u5,u0,w0}
% inner-caps (3): {u1,w0,w1}, {u3,w1,w2}, {u5,w2,w0}
\coordinate (d01) at (barycentric cs:u0=1,u1=1,w0=1);
\coordinate (d12) at (barycentric cs:u1=1,u2=1,w1=1);
\coordinate (d23) at (barycentric cs:u2=1,u3=1,w1=1);
\coordinate (d34) at (barycentric cs:u3=1,u4=1,w2=1);
\coordinate (d45) at (barycentric cs:u4=1,u5=1,w2=1);
\coordinate (d50) at (barycentric cs:u5=1,u0=1,w0=1);
\coordinate (i1) at (barycentric cs:u1=1,w0=1,w1=1);
\coordinate (i3) at (barycentric cs:u3=1,w1=1,w2=1);
\coordinate (i5) at (barycentric cs:u5=1,w2=1,w0=1);
\foreach \p in {d01,d12,d23,d34,d45,d50,i1,i3,i5} {
\node[circle, fill=blue!70!black, inner sep=1.5pt] at (\p) {};
} }
% Highlight one bridge-like annular interior edge — between two inner vertices % Dual cycle edges (crossing each spoke once)
% (For illustration we use the "bridge" between inner i=0 and i=3) \draw[blue!70!black, very thick] (d01) -- (i1);
\draw[gray, dashed, thick] (ui0) to[bend right=15] (ui3); \draw[blue!70!black, very thick] (i1) -- (d12);
% Dual: 12 annular triangles → 12 dual vertices arranged between \draw[blue!70!black, very thick] (d12) -- (d23);
\foreach \i in {0,...,5} { \draw[blue!70!black, very thick] (d23) -- (i3);
\pgfmathsetmacro{\ango}{60*\i + 90 - 15} \draw[blue!70!black, very thick] (i3) -- (d34);
\pgfmathsetmacro{\rmido}{0.5*\Rout + 0.5*\Rin} \draw[blue!70!black, very thick] (d34) -- (d45);
\node[circle, fill=blue!70!black, inner sep=1.4pt] (do\i) at (\ango:\rmido) {}; \draw[blue!70!black, very thick] (d45) -- (i5);
\pgfmathsetmacro{\angi}{60*\i + 90 + 15} \draw[blue!70!black, very thick] (i5) -- (d50);
\node[circle, fill=blue!70!black, inner sep=1.4pt] (di\i) at (\angi:\rmido) {}; \draw[blue!70!black, very thick] (d50) -- (d01);
}
% 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 % Labels
\node[red] at (0, \Rout + 0.35) {\small $B_{\mathrm{out}}$}; \node[red] at (0, \Rout + 0.4) {\small $B_{\mathrm{out}}$ (hexagon)};
\node[red!60!white] at (0, -\Rin + 0.15) {\small $B_{\mathrm{in}}$}; \node[red!55!white] at (\Rin + 0.85, -0.6) {\small $B_{\mathrm{in}}$ (triangle)};
\node[blue!70!black] at (\Rout + 0.85, 0.55) {\small Hamilton walk}; \node[blue!70!black] at (-\Rout - 1.1, 0.4) {\small dual cycle};
\node[blue!70!black] at (\Rout + 0.85, 0.25) {\small + non-crossing}; \node[blue!70!black] at (-\Rout - 1.1, 0.1) {\small $\Gamma \cong C_9$};
\node[blue!70!black] at (\Rout + 0.85, -0.05) {\small chord}; \node[gray] at (\Rout + 0.7, 1.45) {\small spokes};
\end{tikzpicture} \end{tikzpicture}
\caption{Case 2 ($R$ = annulus) with a single ``bridge''-style \caption{Case 2 ($R$ = annulus, spoke-only). Outer boundary
chord. Outer boundary $B_{\mathrm{out}}$ and inner boundary $B_{\mathrm{out}}$ a hexagon (red); inner boundary $B_{\mathrm{in}}$
$B_{\mathrm{in}}$ are concentric hexagons (red). The annular a triangle (light red); $V(O) = V(B_{\mathrm{in}})$ with no chord
region is triangulated by spokes (grey) and one extra interior of $O$, so the triangulation is built purely from spokes (grey)
annular edge between two inner vertices (dashed grey). The between outer and inner vertices. Nine annular triangles (six
inner dual $\Gamma$ (blue) consists of $12$ dual vertices at the ``outer-cap'' triangles with one inner-vertex apex, three
$12$ annular face centroids, connected as a Hamilton cycle around ``inner-cap'' triangles with one outer-vertex apex) tile the
the annulus, plus one chord (dashed blue) corresponding to the annulus. Each blue dot is the centroid of an annular triangle;
extra interior edge. All $12$ vertices lie on the outer face of adjacent dots are joined whenever the two corresponding triangles
the chord-augmented cycle, so $\Gamma$ is outerplanar.} share a spoke. The resulting inner dual $\Gamma$ is the cycle
$C_9$, manifestly outerplanar. For a tire graph with a bridge in
$O$, an additional non-crossing chord appears in $\Gamma$ (see
Remark~\ref{rem:bridge-case-theta}).}
\label{fig:inner-dual-annulus-case} \label{fig:inner-dual-annulus-case}
\end{figure} \end{figure}