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coloring_nested_tire_graphs: redraw Figure 4 with barbell O (real chord)

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Replaces the spoke-only Figure 4 with a true barbell example:

Setup:
  - B_out: hexagon u_0..u_5 (red).
  - O = barbell: triangle {a_1, a_2, a_3} + triangle {b_1, b_2, b_3}
    + bridge a_3-b_1 (light red).
  - 14 spokes triangulate the annulus into 14 annular triangles:
    6 outer-cap + 6 inner-cap + 2 bridge-cap.

Dual placement is precise:
  - All 14 blue dots at exact triangle centroids (via TikZ
    barycentric cs).
  - 13 edges of the Hamilton cycle wrap around the annulus
    crossing each spoke.
  - The bridge dual edge connects the two bridge-cap triangles
    directly (dashed blue chord across the cycle).

Resulting Γ ≅ Θ(1, 7, 7): Hamilton cycle of length 14 with a
single length-1 chord. Outerplanar (the length-1 chord has no
internal degree-2 vertex, so no K_{2,3} minor).

This now properly demonstrates the chord arising from a real
bridge, exactly as the theorem and Remark 1.14 describe.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 02:24:46 -04:00
parent 44d0ed12b0
commit adafecc75b
4 changed files with 98 additions and 81 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\newlabel{fig:inner-dual-disk-case}{{3}{8}} \newlabel{fig:inner-dual-disk-case}{{3}{8}}
\citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals}
\citation{bauerfeld-nested-tire-duals} \citation{bauerfeld-nested-tire-duals}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\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.}}{9}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{4}{9}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
\newlabel{rem:bridge-case-theta}{{1.14}{9}}
\bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-depth}{1}
\bibcite{bauerfeld-nested-tire-duals}{2} \bibcite{bauerfeld-nested-tire-duals}{2}
\newlabel{tocindent-1}{0pt} \newlabel{tocindent-1}{0pt}
@@ -28,9 +32,5 @@
\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, 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 } \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{4}{9}}
\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
\newlabel{rem:bridge-case-theta}{{1.14}{9}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
\gdef \@abspage@last{10} \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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@@ -637,80 +637,97 @@ making $\Gamma$ outerplanar. $\square$
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\begin{tikzpicture}[scale=1.3] \begin{tikzpicture}[scale=1.25]
\def\Rout{2.0}
\def\Rin{0.8}
% Outer hexagon vertices u_i at angles 90, 30, -30, -90, -150, 150 % Outer hexagon vertices u_i at angles 90, 30, -30, -90, -150, 150
\foreach \i in {0,...,5} { \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]90:\scriptsize $u_0$}] (u0) at (0, 2.5) {};
\pgfmathsetmacro{\ang}{90 - 60*\i} \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]30:\scriptsize $u_1$}] (u1) at (2.17, 1.25) {};
\node[circle, fill=black, inner sep=1.3pt, label={\ang:\scriptsize $u_\i$}] (u\i) at (\ang:\Rout) {}; \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]-30:\scriptsize $u_2$}] (u2) at (2.17, -1.25) {};
} \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]-90:\scriptsize $u_3$}] (u3) at (0, -2.5) {};
% Inner triangle vertices w_0 at 60, w_1 at -60, w_2 at 180 \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]-150:\scriptsize $u_4$}](u4) at (-2.17,-1.25) {};
\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=1.2pt, label={[label distance=-1pt]150:\scriptsize $u_5$}] (u5) at (-2.17, 1.25) {};
\node[circle, fill=black, inner sep=1.3pt, label={[label distance=-1pt]-60:\scriptsize $w_1$}] (w1) at (-60:\Rin) {}; % Barbell vertices
\node[circle, fill=black, inner sep=1.3pt, label={[label distance=-1pt]180:\scriptsize $w_2$}] (w2) at (180:\Rin) {}; \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]180:\scriptsize $a_1$}] (a1) at (-0.9, 0.7) {};
% Outer boundary cycle (red) \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]180:\scriptsize $a_2$}] (a2) at (-0.9, -0.7) {};
\foreach \i in {0,...,5} { \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]90:\scriptsize $a_3$}] (a3) at (-0.25, 0) {};
\pgfmathtruncatemacro{\j}{mod(\i+1,6)} \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]90:\scriptsize $b_1$}] (b1) at (0.25, 0) {};
\draw[red, thick] (u\i) -- (u\j); \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]0:\scriptsize $b_2$}] (b2) at (0.9, 0.7) {};
} \node[circle, fill=black, inner sep=1.2pt, label={[label distance=-1pt]0:\scriptsize $b_3$}] (b3) at (0.9, -0.7) {};
% Inner boundary cycle (light red) % B_out edges (red)
\draw[red!55!white, thick] (w0) -- (w1) -- (w2) -- (w0); \draw[red, thick] (u0) -- (u1) -- (u2) -- (u3) -- (u4) -- (u5) -- (u0);
% O edges: left triangle, right triangle, bridge (light red)
\draw[red!55!white, thick] (a1) -- (a2) -- (a3) -- (a1);
\draw[red!55!white, thick] (b1) -- (b2) -- (b3) -- (b1);
\draw[red!55!white, thick] (a3) -- (b1);
% Spokes (gray) % Spokes (gray)
\draw[gray] (u0) -- (w0); \foreach \p/\q in {u0/a1, u0/b2, u0/a3, u0/b1,
\draw[gray] (u1) -- (w0); u1/b2, u1/b3,
\draw[gray] (u1) -- (w1); u2/b3,
\draw[gray] (u2) -- (w1); u3/a2, u3/b3, u3/a3, u3/b1,
\draw[gray] (u3) -- (w1); u4/a2,
\draw[gray] (u3) -- (w2); u5/a1, u5/a2} {
\draw[gray] (u4) -- (w2); \draw[gray, thin] (\p) -- (\q);
\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) {};
} }
% Dual cycle edges (crossing each spoke once) % Dual centroids -- 14 annular triangles
\draw[blue!70!black, very thick] (d01) -- (i1); \coordinate (T1) at (barycentric cs:u0=1,u1=1,b2=1); % outer cap u0-u1
\draw[blue!70!black, very thick] (i1) -- (d12); \coordinate (T2) at (barycentric cs:u1=1,u2=1,b3=1); % outer cap u1-u2
\draw[blue!70!black, very thick] (d12) -- (d23); \coordinate (T3) at (barycentric cs:u2=1,u3=1,b3=1); % outer cap u2-u3
\draw[blue!70!black, very thick] (d23) -- (i3); \coordinate (T4) at (barycentric cs:u3=1,u4=1,a2=1); % outer cap u3-u4
\draw[blue!70!black, very thick] (i3) -- (d34); \coordinate (T5) at (barycentric cs:u4=1,u5=1,a2=1); % outer cap u4-u5
\draw[blue!70!black, very thick] (d34) -- (d45); \coordinate (T6) at (barycentric cs:u5=1,u0=1,a1=1); % outer cap u5-u0
\draw[blue!70!black, very thick] (d45) -- (i5); \coordinate (I1) at (barycentric cs:u5=1,a1=1,a2=1); % inner cap a1-a2
\draw[blue!70!black, very thick] (i5) -- (d50); \coordinate (I2) at (barycentric cs:u3=1,a2=1,a3=1); % inner cap a2-a3
\draw[blue!70!black, very thick] (d50) -- (d01); \coordinate (I3) at (barycentric cs:u0=1,a1=1,a3=1); % inner cap a1-a3
\coordinate (I4) at (barycentric cs:u0=1,b1=1,b2=1); % inner cap b1-b2
\coordinate (I5) at (barycentric cs:u1=1,b2=1,b3=1); % inner cap b2-b3
\coordinate (I6) at (barycentric cs:u3=1,b1=1,b3=1); % inner cap b1-b3
\coordinate (Bup) at (barycentric cs:u0=1,a3=1,b1=1); % bridge cap upper
\coordinate (Bdn) at (barycentric cs:u3=1,a3=1,b1=1); % bridge cap lower
\foreach \p in {T1,T2,T3,T4,T5,T6,I1,I2,I3,I4,I5,I6,Bup,Bdn} {
\node[circle, fill=blue!70!black, inner sep=1.4pt] at (\p) {};
}
% Hamilton cycle of length 14 (going clockwise from Bup)
\draw[blue!70!black, very thick] (Bup) -- (I4); % share u0-b1
\draw[blue!70!black, very thick] (I4) -- (T1); % share u0-b2
\draw[blue!70!black, very thick] (T1) -- (I5); % share u1-b2
\draw[blue!70!black, very thick] (I5) -- (T2); % share u1-b3
\draw[blue!70!black, very thick] (T2) -- (T3); % share u2-b3
\draw[blue!70!black, very thick] (T3) -- (I6); % share u3-b3
\draw[blue!70!black, very thick] (I6) -- (Bdn); % share u3-b1
\draw[blue!70!black, very thick] (Bdn) -- (I2); % share u3-a3
\draw[blue!70!black, very thick] (I2) -- (T4); % share u3-a2
\draw[blue!70!black, very thick] (T4) -- (T5); % share u4-a2
\draw[blue!70!black, very thick] (T5) -- (I1); % share u5-a2
\draw[blue!70!black, very thick] (I1) -- (T6); % share u5-a1
\draw[blue!70!black, very thick] (T6) -- (I3); % share u0-a1
\draw[blue!70!black, very thick] (I3) -- (Bup); % share u0-a3
% Chord: bridge dual edge
\draw[blue!70!black, very thick, dashed] (Bup) -- (Bdn);
% Labels % Labels
\node[red] at (0, \Rout + 0.4) {\small $B_{\mathrm{out}}$ (hexagon)}; \node[red] at (0, 3.05) {\small $B_{\mathrm{out}}$ (hexagon)};
\node[red!55!white] at (\Rin + 0.85, -0.6) {\small $B_{\mathrm{in}}$ (triangle)}; \node[red!55!white] at (2.85, 0.0) {\small barbell $O$};
\node[blue!70!black] at (-\Rout - 1.1, 0.4) {\small dual cycle}; \node[blue!70!black] at (-3.15, 0.5) {\small Hamilton cycle};
\node[blue!70!black] at (-\Rout - 1.1, 0.1) {\small $\Gamma \cong C_9$}; \node[blue!70!black] at (-3.15, 0.18) {\small (length 14)};
\node[gray] at (\Rout + 0.7, 1.45) {\small spokes}; \node[blue!70!black] at (-3.15, -0.4) {\small chord = bridge};
\node[blue!70!black] at (-3.15, -0.7) {\small dual edge};
\draw[->, blue!70!black, thin] (-2.5, -0.4) -- (-0.15, 0);
\end{tikzpicture} \end{tikzpicture}
\caption{Case 2 ($R$ = annulus, spoke-only). Outer boundary \caption{Case 2 ($R$ = annulus) with $O$ a barbell.
$B_{\mathrm{out}}$ a hexagon (red); inner boundary $B_{\mathrm{in}}$ $B_{\mathrm{out}}$ is the outer hexagon (red); $O$ has two
a triangle (light red); $V(O) = V(B_{\mathrm{in}})$ with no chord triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by
of $O$, so the triangulation is built purely from spokes (grey) the bridge $a_3\text{--}b_1$ (all light red). The annulus is
between outer and inner vertices. Nine annular triangles (six triangulated by $14$ annular triangles: $6$ ``outer-cap''
``outer-cap'' triangles with one inner-vertex apex, three triangles (one per outer edge), $6$ ``inner-cap'' triangles (one
``inner-cap'' triangles with one outer-vertex apex) tile the per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles
annulus. Each blue dot is the centroid of an annular triangle; $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the
adjacent dots are joined whenever the two corresponding triangles bridge. Each blue dot sits at the centroid of an annular
share a spoke. The resulting inner dual $\Gamma$ is the cycle triangle; blue edges connect dual vertices whose triangles share
$C_9$, manifestly outerplanar. For a tire graph with a bridge in an interior annular edge (spoke or bridge). The two bridge-cap
$O$, an additional non-crossing chord appears in $\Gamma$ (see vertices have $\Gamma$-degree $3$ (their triangles have no
Remark~\ref{rem:bridge-case-theta}).} 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.}
\label{fig:inner-dual-annulus-case} \label{fig:inner-dual-annulus-case}
\end{figure} \end{figure}