diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index 77575ce..edb0823 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -8,7 +8,7 @@ \citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \newlabel{def:partial-tire-dual}{{1.1}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } \newlabel{fig:partial-tire-dual-example}{{1}{2}} \newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} \citation{bauerfeld-nested-tires} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index 4d55b1f..f0740d3 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_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 00:58 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:01 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -258,7 +258,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (9 pages, 605784 bytes). +Output written on paper.pdf (9 pages, 605799 bytes). PDF statistics: 151 PDF objects out of 1000 (max. 8388607) 89 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index ad28826..a74bb72 100644 Binary files a/papers/coloring_nested_tire_dual_graphs/paper.pdf and b/papers/coloring_nested_tire_dual_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_dual_graphs/paper.tex b/papers/coloring_nested_tire_dual_graphs/paper.tex index 97746a8..870fa51 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -100,8 +100,8 @@ and $G$ has $2n - 4$ triangular faces. Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in the sense of \cite[Definition~1.5]{bauerfeld-nested-tires}, and let $F_{\mathrm{ann}}$ -denote the set of triangular faces of $T$ in the closed annular region -between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial +denote the set of triangular faces of $T$ in the tire tread +(the closed region between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$). The \emph{partial tire dual} of $T$, written $D(T)$, is the graph defined as follows. \emph{Vertices.} @@ -145,7 +145,7 @@ tire dual} of $T$, written $D(T)$, is the graph defined as follows. diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each -boundary edge of the annular region (either of $B_{\mathrm{out}}$ or +boundary edge of the tire tread (either of $B_{\mathrm{out}}$ or of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of @@ -198,13 +198,13 @@ $\{x, y, z\}$ with $x \in V(B_{\mathrm{out}})$, $y \in V(B_{\mathrm{in}})$, and $z$ also in $V(B_{\mathrm{out}}) \cup V(B_{\mathrm{in}})$. Of its three edges, the one between the two same-side vertices ($x$-$z$ if both on $B_{\mathrm{out}}$, or $y$-$z$ if both on -$B_{\mathrm{in}}$) is a boundary edge of the annular region; the +$B_{\mathrm{in}}$) is a boundary edge of the tire tread; the other two edges are spokes. So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes shared with adjacent annular faces) and one leaf. The induced subgraph on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the -connectedness of the annular region this forces it to be a single +connectedness of the tire tread this forces it to be a single cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + m$, and there are also $n + m$ leaves attached one-per-cycle-vertex. \end{proof} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 476f6a6..0050bb9 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 00:57 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:01 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -202,7 +202,7 @@ Package pdftex.def Info: fig_dual_depth.png used on input line 117. File: fig_tire_example.png Graphic file (type png) -Package pdftex.def Info: fig_tire_example.png used on input line 171. +Package pdftex.def Info: fig_tire_example.png used on input line 172. (pdftex.def) Requested size: 280.79956pt x 188.56097pt. [3 <./fig_tire_example.png>] (./paper.aux) ) Here is how much of TeX's memory you used: @@ -212,7 +212,7 @@ Here is how much of TeX's memory you used: 21056 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,625b,276s stack positions out of 10000i,1000n,20000p,200000b,200000s + 69i,7n,76p,625b,278s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on paper.pdf (3 pages, 456808 bytes). +Output written on paper.pdf (3 pages, 457250 bytes). PDF statistics: 94 PDF objects out of 1000 (max. 8388607) 54 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 6035c95..d5f76eb 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index b7aabd2..f1fc28c 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -50,9 +50,9 @@ triangulations. A \emph{level source} of a triangulation $G$ induces a BFS layering of $G$, which in turn endows the inner planar dual $G'$ with a \emph{dual depth} grading. We isolate the basic object of study --- the \emph{tire graph} $T$, a plane graph -whose outer and inner boundaries bound an annular region -triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$ --- and -record its face/edge counts. +whose outer and inner boundaries bound the \emph{tire tread} $R$, +a closed region triangulated by the \emph{annular edges} +$E_{\mathrm{ann}}$ --- and record its face/edge counts. \end{abstract} \maketitle @@ -150,20 +150,21 @@ The vertex and edge sets of $T$ are E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, \] where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the -property that, in the plane embedding of $T$, the closed planar region -$R$ bounded externally by $B_{\mathrm{out}}$ and internally by -$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose -union is $R$. +property that, in the plane embedding of $T$, the closed planar +region $R$ bounded externally by $B_{\mathrm{out}}$ and internally +by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ +whose union is $R$. We call $R$ the \emph{tire tread} of $T$. When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected, -$R$ is a closed annulus. More generally, $R$ is a closed planar -region that may fail to be a $2$-manifold at cut-vertices of $O$ (where -two ``lobes'' of the depth-$d$ region meet at a single vertex); the -inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that -visits the cut-vertex multiple times. The relaxed definition -accommodates outerplanar inner graphs with bridges, cut-vertices, or -multiple connected components. When either boundary is degenerate, -$R$ is a closed disk with that vertex as apex. +the tread is a closed annulus. More generally, $R$ is a closed +planar region that may fail to be a $2$-manifold at cut-vertices of +$O$ (where two ``lobes'' of the depth-$d$ region meet at a single +vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple +closed walk that visits the cut-vertex multiple times. The relaxed +definition accommodates outerplanar inner graphs with bridges, +cut-vertices, or multiple connected components. When either +boundary is degenerate, the tread is a closed disk with that vertex +as apex. \end{definition} \begin{figure}[h] @@ -182,8 +183,8 @@ triangular faces.} \begin{remark} \label{rem:tire-counts} Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By -Euler's formula on the annular (resp.\ disk) region $R$, the tire graph -has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ +Euler's formula on the tire tread $R$, the tire graph has $m + k$ +triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ annular edges when neither boundary is degenerate; when exactly one boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.