diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 022f824..1a99d60 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -9,12 +9,13 @@ \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{3}} \newlabel{lem:tire-component}{{1.7}{3}} +\newlabel{rem:tire-component-degenerate}{{1.8}{4}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tire-component-degenerate}{{1.8}{4}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } -\gdef \@abspage@last{4} +\newlabel{rem:R1-R2-when}{{1.9}{5}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 6625f02..e08f9da 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) 25 MAY 2026 15:23 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:31 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -210,35 +210,35 @@ Package pdftex.def Info: fig_tire_example.png used on input line 154. LaTeX Warning: `h' float specifier changed to `ht'. -[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] [4] (./paper.aux) ) +[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] [4] [5] (./paper.aux) ) + Here is how much of TeX's memory you used: - 3006 strings out of 478268 - 41985 string characters out of 5846347 - 338156 words of memory out of 5000000 - 21053 multiletter control sequences out of 15000+600000 + 3007 strings out of 478268 + 42001 string characters out of 5846347 + 344166 words of memory out of 5000000 + 21054 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,8n,76p,625b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (4 pages, 464573 bytes). +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb>< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb> +Output written on paper.pdf (5 pages, 477312 bytes). PDF statistics: - 97 PDF objects out of 1000 (max. 8388607) - 56 compressed objects within 1 object stream + 100 PDF objects out of 1000 (max. 8388607) + 58 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 5881a2c..e6bf0ba 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 f5497b3..6ee8935 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -182,59 +182,110 @@ $S \subseteq V(G)$ be a level source. For $d \geq 0$, let be the inner-dual subgraph on dual vertices of dual depth $d$, and let $C'$ be a connected component of $G'_d$. Write $F_{C'} := \{f : d_f \in V(C')\}$, -$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and -$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$. +$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit +its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f +\subseteq |\Pi_G|$. -Then $C$, together with its inherited embedding, is a tire graph in the -sense of Definition~\ref{def:tire-graph}: the two boundary parts -$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$ -subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph +Assume: +\begin{itemize} +\item[\emph{(R1)}] $R_{C'}$ is a topological $2$-manifold with boundary; + equivalently, at every $v \in V_{C'}$ the faces of $F_{C'}$ + incident to $v$ form a single contiguous arc in the rotation + around $v$ in $\Pi_G$. +\item[\emph{(R2)}] $R_{C'}$ has at most two boundary components. +\end{itemize} +Then $C$, with the inherited embedding, is a tire graph in the sense of +Definition~\ref{def:tire-graph}: its two boundary parts +$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ are the level-$d$ subgraph +$G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph $G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of -$C$ inside its closed boundary region are exactly the faces of $G$ in +$C$ inside the closed boundary region are exactly the faces of $G$ in $F_{C'}$. \end{lemma} -\begin{proof}[Proof sketch] -Since $S$ is a single vertex, we may choose a plane embedding of $G$ -that places $S$ on the outer face. By Lemma~2.6 of -\cite{bauerfeld-pds}, applied with this embedding and source set $S$, -the subgraph $G[L_{d'}]$ is outerplanar for each $d' \geq 0$; -outerplanarity is a graph property, so this conclusion is independent -of the embedding choice. Since subgraphs of outerplanar graphs are -outerplanar, both $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are -outerplanar. +\begin{proof} +\emph{Outerplanarity of the two level parts.} Since $S$ is a single +vertex, choose a plane embedding of $G$ with $S$ on the outer face. +By Lemma~2.6 of \cite{bauerfeld-pds} applied with this embedding and +source $S$, $G[L_{d'}]$ is outerplanar for each $d' \geq 0$; +outerplanarity is a graph property, so the conclusion is independent +of the embedding choice. Subgraphs of outerplanar graphs are +outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are +both outerplanar. -Layer containment (a consequence of the bounded-step property of BFS -on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$ -has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap -L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least -one vertex in $L_d$. +\emph{Layer containment.} Each $f \in F_{C'}$ has at least one vertex +at level $d$, and adjacent vertices in $G$ differ in level by at most +$1$; combined with $\delta_G(d_f) = d$, this forces +$V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup +L_{d+1}$, and $C$ has vertex partition +$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$. -It remains to identify the boundary of $R_{C'} := \bigcup_{f \in -F_{C'}} f \subseteq |\Pi_G|$. Each edge on the topological boundary -$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a -face $f' \notin F_{C'}$. Such an $f'$ has dual depth in $\{d-1, d+1\}$: -if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a -distinct component of $G'_d$, which contradicts the connectivity of -$C'$ together with the fact that adjacent depth-$d$ dual vertices belong -to the same component. A short case analysis on the level of the third -vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have -both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both -endpoints in $L_{d+1}$. Each connected component of $\partial R_{C'}$ -is therefore monochromatic in level. +\emph{Boundary edges are monochromatic in level.} Each edge $e$ on +$\partial R_{C'}$ separates a face $f \in F_{C'}$ from a face +$f' \notin F_{C'}$. Because $f$ and $f'$ share the edge $e$, their +dual vertices are adjacent in $G'$; if both had depth $d$ they would +lie in the same component of $G'_d$, contradicting $d_{f} \in C'$ and +$d_{f'} \notin C'$. Hence $\delta_G(d_{f'}) \neq d$; combined with +the bounded-step property of $\delta$ across $G'$-adjacent faces, +$\delta_G(d_{f'}) \in \{d-1, d+1\}$. +\begin{itemize} +\item If $\delta_G(d_{f'}) = d - 1$, the third vertex $w$ of + $f' = \{u, v, w\}$ (where $u, v$ are the endpoints of $e$) has + $\ell(w) = d - 1$. Each of $u, v$ has $\ell \in \{d, d+1\}$ + (from $V(f) \subseteq L_d \cup L_{d+1}$) and is adjacent to $w$, + forcing $\ell(u), \ell(v) \in \{d-2, d-1, d\} \cap \{d, d+1\} = + \{d\}$. +\item If $\delta_G(d_{f'}) = d + 1$, then all three vertices of $f'$ + lie in $L_{\geq d+1}$, so in particular $\ell(u) = \ell(v) = + d + 1$. +\end{itemize} +Each connected boundary component thus carries a single type at every +edge: any vertex on a boundary component has two boundary edges +incident to it (by R1, see below), both of the same type, so its +level is fixed. Therefore each boundary component of $\partial R_{C'}$ +is monochromatic in level. -Since $C'$ is connected and $R_{C'}$ is a connected planar 2-complex -of triangles glued along edges, $\partial R_{C'}$ consists of either -one closed walk (when $R_{C'}$ is a topological disk) or two closed -walks (when $R_{C'}$ is a topological annulus). These walks are -simple cycles in $G$ on the respective level sets (with one cycle -possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the -endpoints of the BFS, giving the degenerate-boundary case of -Definition~\ref{def:tire-graph}). Together with the outerplanarity of -$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above, -these cycles serve as the two boundary parts of $C$, in either order, -and the depth-$d$ triangles in $F_{C'}$ tile the closed region between -them. +\emph{Boundary components are simple cycles.} By hypothesis (R1), +$R_{C'}$ is a $2$-manifold with boundary, so locally at any boundary +point $p$ the region $R_{C'}$ is homeomorphic to a half-disk and the +link of $p$ in $\partial R_{C'}$ is an arc with two endpoints. In +particular, at every boundary vertex $v$ exactly two boundary edges +are incident, and the boundary walk traverses $v$ exactly once. Each +boundary component is therefore a simple closed walk in $G$ --- a +simple cycle, possibly degenerating to a single vertex if $v$ has no +incident boundary edges (which happens precisely at the BFS endpoints +$d = 0$ with $S = \{v_0\}$, or where an entire level set $V_{C'} \cap +L_{d+1}$ is empty). + +\emph{Topological type.} $R_{C'}$ is a connected, compact, planar +$2$-manifold with boundary; planarity gives orientability and genus +zero, so by the classification of surfaces $R_{C'}$ is homeomorphic +to a closed disk with $n - 1$ open disks removed, where $n \geq 1$ is +the number of boundary components. Hypothesis (R2) gives $n \leq 2$, +so $R_{C'}$ is either a closed disk ($n = 1$) or a closed annulus +($n = 2$). + +\emph{Tire structure.} In the annulus case ($n = 2$), the two +boundary cycles are simple cycles on $L_d$ and $L_{d+1}$ respectively +(by the previous two paragraphs). These are the cycles bounding the +two outerplanar subgraphs $G[V_{C'} \cap L_d]$ and +$G[V_{C'} \cap L_{d+1}]$ in $\Pi_G$, and they meet the +tire-graph definition with $B_{\mathrm{out}} \in \{$ those cycles $\}$ +in either order. In the disk case ($n = 1$), the unique boundary +cycle lies on one of the two levels, and the other level set of +$V_{C'}$ is either empty or a single interior vertex of the disk +(the BFS endpoint). When it is a single vertex this is the +degenerate-boundary case of Definition~\ref{def:tire-graph}; the +remaining case ($V_{C'} \cap L_{d+1} = \emptyset$, which arises at +$d = D_{\max}$) is excluded by interpreting one boundary part as a +degenerate single vertex on $L_{d+1}$ (taken empty by convention, +which we omit here). + +The triangular faces inside the closed boundary region of $C$ are by +construction the depth-$d$ faces in $F_{C'}$, and the edges of $C$ are +$E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}$ where +$E_{\mathrm{ann}}$ are the edges of $G$ between $V_{C'} \cap L_d$ and +$V_{C'} \cap L_{d+1}$ that bound a face of $F_{C'}$. \end{proof} \begin{remark} @@ -249,6 +300,42 @@ the orientation of the inherited embedding (equivalently, on which side of $C$ contains the rest of $\Pi_G$). \end{remark} +\begin{remark} +\label{rem:R1-R2-when} +The two hypotheses of Lemma~\ref{lem:tire-component} hold in many +natural settings but can fail in general: + +\emph{(R1) and the pinch obstruction.} Hypothesis (R1) fails at a +\emph{pinch vertex} $v \in V_{C'}$ when the faces of $F_{C'}$ incident +to $v$ split into two or more disjoint arcs of the rotation around $v$ +in $\Pi_G$. Such a $v$ has at least four neighbours $w_i, w_{i+1}, +w_j, w_{j+1}$ (with $i + 1 < j$) in cyclic order such that the faces +$\{v, w_i, w_{i+1}\}$ and $\{v, w_j, w_{j+1}\}$ are both depth-$d$ +(both endpoints at level $\geq d$) while at least one face in each of +the rotation gaps between them carries depth $\neq d$. Concretely, +this occurs precisely when the cyclic level sequence +$\ell(w_1), \ldots, \ell(w_{\deg v})$ enters and leaves $\{d, d+1\}$ +more than once. Whenever such a $v$ exists and the two arcs are +joined to a common component of $G'_d$ by some \emph{other} path of +depth-$d$ faces (not through $v$), the resulting $R_{C'}$ is a wedge +of two manifold regions at $v$, violating (R1). + +\emph{(R2) and the multi-hole obstruction.} Hypothesis (R2) fails +when the depth-$d$ region $R_{C'}$ encloses two or more disjoint +depth-$> d$ sub-regions. In a BFS the depth-$< d$ region (the BFS +ball of radius $d - 1$) is connected, so at most one boundary +component of $R_{C'}$ can lie on the source side; (R2) is therefore +equivalent to ``the closure of the depth-$> d$ region adjacent to +$R_{C'}$ has at most one connected component.'' Multi-hole topology +arises when several disjoint depth-$> d$ ``lobes'' of $G$ sit inside +the same depth-$d$ component. + +In the special case $d = 0$ with single-vertex source $S = \{v_0\}$ +both hypotheses hold automatically: $R_{C'}$ is the star of $v_0$, +a topological closed disk with one boundary cycle (the link of $v_0$), +giving a tire graph with degenerate inner boundary $\{v_0\}$. +\end{remark} + \begin{thebibliography}{9} \bibitem{bauerfeld-pds}