diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index edb0823..a0ae3e6 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -21,20 +21,23 @@ \citation{bauerfeld-nested-tires} \citation{bauerfeld-depth} \citation{bauerfeld-nested-tires} -\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}} +\newlabel{thm:tread-partition}{{1.5}{6}} +\newlabel{rem:tire-component-degenerate}{{1.6}{6}} \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} -\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}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.7}{7}} +\newlabel{prop:edge-vertex-bijection}{{1.8}{7}} +\newlabel{rem:edge-vertex-corollary}{{1.9}{7}} +\newlabel{def:tire-annular-subgraph}{{1.10}{7}} +\newlabel{def:tire-annular-face-connector}{{1.11}{8}} +\newlabel{def:spokes}{{1.12}{8}} +\newlabel{rem:facial-dual-spoke-only}{{1.13}{8}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{8}{}\protected@file@percent } \newlabel{sec:latin-conjecture}{{2}{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 $u_4, u_5$).}}{9}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{3}{9}} +\newlabel{conj:latin}{{2.1}{9}} +\newlabel{conj:chain-latin}{{2.2}{9}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} @@ -42,7 +45,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\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} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } +\gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index f0740d3..88a767b 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 01:01 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:09 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -209,14 +209,18 @@ LaTeX Warning: `h' float specifier changed to `ht'. 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PDF statistics: - 151 PDF objects out of 1000 (max. 8388607) - 89 compressed objects within 1 object stream + 159 PDF objects out of 1000 (max. 8388607) + 94 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 16 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index a74bb72..28c8f73 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 870fa51..2ecc0b4 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -388,6 +388,59 @@ between $V_{C'} \cap L_d$ and $V_{C'} \cap L_{d+1}$ that bound a face of $F_{C'}$. \end{proof} +\begin{theorem}[Tire treads partition the bounded faces] +\label{thm:tread-partition} +Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and +let $S \subseteq V(G)$ be a level source lying on the outer face. +For each $d \ge 0$ and each connected component $C'$ of $G'_d$, let +$T^{(d, C')}$ denote the tire graph supplied by +Lemma~\ref{lem:tire-component}, with tire tread +$R_{C'} \subseteq |\Pi_G|$. Then the collection of treads +\[ + \mathcal{R}(G, S) \;:=\; + \bigl\{\, R_{C'} \,:\, d \ge 0,\; + C' \text{ a connected component of } G'_d \,\bigr\} +\] +partitions the bounded part of $|\Pi_G|$: +\begin{enumerate} + \item[(i)] every bounded face $f$ of $G$ is contained in exactly + one tread $R_{C'} \in \mathcal{R}(G, S)$; + \item[(ii)] distinct treads in $\mathcal{R}(G, S)$ have disjoint + interiors and may share only boundary edges or vertices. +\end{enumerate} +\end{theorem} + +\begin{proof} +\emph{Existence and uniqueness.} Each bounded face $f \in F(G)$ +has a uniquely-defined dual depth $\delta_G(d_f) \in \mathbb{Z}_{\ge +0}$, so the dual vertex $d_f$ lies in $G'_d$ for $d = +\delta_G(d_f)$ and in no other $G'_{d'}$. Within $G'_d$, the +vertex $d_f$ belongs to exactly one connected component $C'$. By +Lemma~\ref{lem:tire-component}, $F_{C'}$ is precisely the set of +faces $f' \in F(G)$ with $d_{f'} \in V(C')$; in particular $f \in +F_{C'}$, hence $f \subseteq R_{C'}$. + +For any other tread $R_{C''} \in \mathcal{R}(G, S)$, the +component $C''$ is either at a different depth $d' \ne d$ (in +which case $F_{C''}$ consists of depth-$d'$ faces and $f \notin +F_{C''}$) or at depth $d$ but a different component $C'' \ne C'$ +(in which case the two components are vertex-disjoint in $G'_d$, +so again $f \notin F_{C''}$). In both cases $f \notin R_{C''}$ +(more precisely, $f$ is not one of the triangular faces of $G$ in +$F_{C''}$, so $f$'s interior is not contained in $R_{C''}$). + +\emph{Disjoint interiors.} Each tread $R_{C'}$ is the union of +its triangular faces $F_{C'} \subseteq F(G)$; distinct treads +correspond to disjoint $F_{C'}$ (by the argument above), and the +interiors of distinct $G$-faces are disjoint. Hence interiors of +distinct treads are disjoint. + +\emph{Coverage.} Conversely, every bounded $f \in F(G)$ has $d_f +\in V(G')$ with some dual depth $d$, and thus lies in $R_{C'}$ +where $C'$ is its component of $G'_d$. So $\bigcup_{R \in +\mathcal{R}(G, S)} R$ contains every bounded face of $G$. +\end{proof} + \begin{remark} \label{rem:tire-component-degenerate} Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be