From df9411d2bd3e3a2dc2847a6a368b6c8c99dc8042 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 00:57:31 -0400 Subject: [PATCH] papers: move Definition 1.7 (partial tire dual) to dual paper MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit REMOVED from coloring_nested_tire_graphs/: - Definition 1.7 (Partial tire dual) - Figure 3 (partial tire dual example) - Figure 4 (partial tire dual bridge case) - fig_partial_tire_dual.png file - fig_partial_tire_dual_bridge.png file - Abstract no longer mentions partial tire dual Foundational paper now ends at Remark 1.6 (tire face/edge counts). Down from 5 to 3 pages. ADDED to coloring_nested_tire_dual_graphs/: - Definition (Partial tire dual) — now numbered 1.1 in this paper - Figure: partial tire dual example - Figure: partial tire dual bridge case - Both PNG figure files Inserted before the structure proposition (former 1.1, now 1.2). Intro citation list removes the bullet for partial tire dual since it's now defined locally. The definition's internal ref to Definition~\ref{def:tire-graph} becomes \cite[Definition~1.5]{bauerfeld-nested-tires}. The two figure captions updated to reference prop:partial-tire-dual-structure locally (instead of citing the companion paper as if it owned the definition). Paper grows from 8 to 9 pages. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 47 +++++---- .../paper.log | 97 ++++++++++-------- .../paper.pdf | Bin 362539 -> 605744 bytes .../paper.tex | 82 ++++++++++++++- .../fig_partial_tire_dual.png | Bin 150413 -> 0 bytes .../fig_partial_tire_dual_bridge.png | Bin 146308 -> 0 bytes papers/coloring_nested_tire_graphs/paper.aux | 16 +-- papers/coloring_nested_tire_graphs/paper.log | 75 ++++++-------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 711472 -> 456808 bytes papers/coloring_nested_tire_graphs/paper.tex | 81 +-------------- 10 files changed, 195 insertions(+), 203 deletions(-) delete mode 100644 papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png delete mode 100644 papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index 4696eed..77575ce 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -7,31 +7,34 @@ \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{prop:partial-tire-dual-structure}{{1.1}{1}} +\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 } +\newlabel{fig:partial-tire-dual-example}{{1}{2}} +\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} \citation{bauerfeld-nested-tires} -\newlabel{prop:no-level-d-pinch}{{1.2}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{3}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-bridge}{{2}{3}} +\newlabel{prop:no-level-d-pinch}{{1.3}{3}} +\citation{bauerfeld-nested-tires} +\newlabel{lem:tire-component}{{1.4}{4}} +\citation{bauerfeld-depth} \citation{bauerfeld-nested-tires} \citation{bauerfeld-depth} \citation{bauerfeld-nested-tires} -\newlabel{lem:tire-component}{{1.3}{3}} -\citation{bauerfeld-depth} -\citation{bauerfeld-nested-tires} -\newlabel{rem:tire-component-degenerate}{{1.4}{4}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.5}{4}} -\newlabel{prop:edge-vertex-bijection}{{1.6}{4}} +\newlabel{rem:tire-component-degenerate}{{1.5}{6}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.6}{6}} +\newlabel{prop:edge-vertex-bijection}{{1.7}{6}} \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} -\newlabel{rem:edge-vertex-corollary}{{1.7}{5}} -\newlabel{def:tire-annular-subgraph}{{1.8}{5}} -\newlabel{def:tire-annular-face-connector}{{1.9}{5}} -\newlabel{def:spokes}{{1.10}{6}} -\newlabel{rem:facial-dual-spoke-only}{{1.11}{6}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{6}{}\protected@file@percent } -\newlabel{sec:latin-conjecture}{{2}{6}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{7}{}\protected@file@percent } -\newlabel{fig:facial-dual-choices}{{1}{7}} -\newlabel{conj:latin}{{2.1}{7}} -\newlabel{conj:chain-latin}{{2.2}{7}} +\newlabel{rem:edge-vertex-corollary}{{1.8}{7}} +\newlabel{def:tire-annular-subgraph}{{1.9}{7}} +\newlabel{def:tire-annular-face-connector}{{1.10}{7}} +\newlabel{def:spokes}{{1.11}{7}} +\newlabel{rem:facial-dual-spoke-only}{{1.12}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{8}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{3}{8}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{8}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{8}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} @@ -39,5 +42,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } -\gdef \@abspage@last{8} +\newlabel{conj:latin}{{2.1}{9}} +\newlabel{conj:chain-latin}{{2.2}{9}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } +\gdef \@abspage@last{9} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index b9e1bdf..bd2c1c9 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 00:54 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,19 +192,31 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2] [3] [4] [5] - -File: notes/fig_facial_dual_choices.png Graphic file (type png) - -Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -495. -(pdftex.def) Requested size: 360.0pt x 143.50418pt. + +File: fig_partial_tire_dual.png Graphic file (type png) + +Package pdftex.def Info: fig_partial_tire_dual.png used on input line 143. +(pdftex.def) Requested size: 280.79956pt x 233.36552pt. + +File: fig_partial_tire_dual_bridge.png Graphic file (type png) + +Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 1 +58. +(pdftex.def) Requested size: 306.0022pt x 204.59406pt. 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z?!F%DsNdw<=qP)NF6%7bV3C12SkKn;S!ic9Rx0Xp9MNS!Cs*-KJjIxojMej+a7SuK x`%auD9PW)eKUOLF>})o!-R>%Xfs&F5^Q8#BK;a{De&0MUCO8TTaYYHZ{{_Sya<%{f diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 770660f..b7aabd2 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -52,9 +52,7 @@ 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 -define its \emph{partial tire dual} $D(T)$, the dual restricted to -$T$'s annular faces together with leaves recording the boundary -edges. +record its face/edge counts. \end{abstract} \maketitle @@ -191,83 +189,6 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. \end{remark} -\begin{definition}[Partial tire dual] -\label{def:partial-tire-dual} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in -the sense of Definition~\ref{def:tire-graph}, 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 -tire dual} of $T$, written $D(T)$, is the graph defined as follows. - -\emph{Vertices.} -\begin{enumerate} - \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an - \emph{interior vertex} $d_f$ of $D(T)$. - \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a - \emph{leaf vertex} $\ell_e^{\mathrm{out}}$. - \item[(V3)] For each occurrence of an edge in the closed walk - $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$), - a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is - $2$-connected each edge appears once; cut-vertices and - bridges of $O$ may cause an edge or vertex to appear more - than once.) -\end{enumerate} - -\emph{Edges.} -\begin{enumerate} - \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both - lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}), - one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where - $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces - incident to $e$. - \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge - $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where - $f \in F_{\mathrm{ann}}$ is the unique annular face incident - to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$. - \item[(E3)] For each occurrence of $e$ on the boundary walk - $B_{\mathrm{in}}$, one edge - $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where - $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$ - on the side of that occurrence. The leaf - $\ell_e^{\mathrm{in}}$ has degree $1$. -\end{enumerate} -\end{definition} - -\begin{figure}[h] -\centering -\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png} -\caption{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 $C_{10} \circ K_1$ analysed in the companion -paper~\cite{bauerfeld-nested-tire-duals}.} -\label{fig:partial-tire-dual-example} -\end{figure} - -\begin{figure}[h] -\centering -\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png} -\caption{Partial tire dual $D(T)$ when the inner outerplanar graph -$O$ has a bridge --- here a non-trivial edge cut connecting two -disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on -$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together -with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing -the bridge disconnects $O$). Because both faces incident to the -bridge are annular triangles, the bridge contributes an -\emph{interior dual edge} (highlighted in red) rather than two -leaves; consequently the interior dual subgraph is no longer the -single $(n+m)$-cycle of the spoke-only case, but a theta -graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident -annular faces) are joined by three internally vertex-disjoint paths -in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves) -and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves, -three for each triangle).} -\label{fig:partial-tire-dual-bridge} -\end{figure} \begin{thebibliography}{9}