From 44d0ed12b066c891d861e3efdd492b1ed0f1d79a Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 02:02:56 -0400 Subject: [PATCH] coloring_nested_tire_graphs: redraw Figure 4 properly MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- papers/coloring_nested_tire_graphs/paper.aux | 4 +- papers/coloring_nested_tire_graphs/paper.log | 68 +++++------ papers/coloring_nested_tire_graphs/paper.pdf | Bin 590548 -> 592331 bytes papers/coloring_nested_tire_graphs/paper.tex | 116 +++++++++++-------- 4 files changed, 101 insertions(+), 87 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 23c4271..c2d70f3 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -28,9 +28,9 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{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{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{9} +\gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index b363c87..b2d103b 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:53 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 02:02 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,43 +511,43 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. 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- \foreach \i in {0,...,5} { - \pgfmathsetmacro{\ang}{60*\i + 90 + 30} - \node[circle, fill=black, inner sep=1.2pt] (ui\i) at (\ang:\Rin) {}; + \pgfmathsetmacro{\ang}{90 - 60*\i} + \node[circle, fill=black, inner sep=1.3pt, label={\ang:\scriptsize $u_\i$}] (u\i) at (\ang:\Rout) {}; } + % 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} { \pgfmathtruncatemacro{\j}{mod(\i+1,6)} - \draw[red, thick] (uo\i) -- (uo\j); - \draw[red!60!white, thick] (ui\i) -- (ui\j); + \draw[red, thick] (u\i) -- (u\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); + % Inner boundary cycle (light red) + \draw[red!55!white, thick] (w0) -- (w1) -- (w2) -- (w0); + % Spokes (gray) + \draw[gray] (u0) -- (w0); + \draw[gray] (u1) -- (w0); + \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 - % (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); + % Dual cycle edges (crossing each spoke once) + \draw[blue!70!black, very thick] (d01) -- (i1); + \draw[blue!70!black, very thick] (i1) -- (d12); + \draw[blue!70!black, very thick] (d12) -- (d23); + \draw[blue!70!black, very thick] (d23) -- (i3); + \draw[blue!70!black, very thick] (i3) -- (d34); + \draw[blue!70!black, very thick] (d34) -- (d45); + \draw[blue!70!black, very thick] (d45) -- (i5); + \draw[blue!70!black, very thick] (i5) -- (d50); + \draw[blue!70!black, very thick] (d50) -- (d01); % 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}; + \node[red] at (0, \Rout + 0.4) {\small $B_{\mathrm{out}}$ (hexagon)}; + \node[red!55!white] at (\Rin + 0.85, -0.6) {\small $B_{\mathrm{in}}$ (triangle)}; + \node[blue!70!black] at (-\Rout - 1.1, 0.4) {\small dual cycle}; + \node[blue!70!black] at (-\Rout - 1.1, 0.1) {\small $\Gamma \cong C_9$}; + \node[gray] at (\Rout + 0.7, 1.45) {\small spokes}; \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.} +\caption{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~\ref{rem:bridge-case-theta}).} \label{fig:inner-dual-annulus-case} \end{figure}