From 1ac80aa5cfc545830aae109770e53953a2e85863 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 15:36:26 -0400 Subject: [PATCH] coloring_nested_tire_graphs: simplify Lemma 1.7 by requiring S on the outer face MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Pins Π_G at the start to be an embedding placing S on the outer face; such an embedding exists for any single-vertex source. This collapses the two-embedding split in the previous proof (one for Lemma 2.6, another for the topological analysis of R_{C'}) into a single embedding throughout, and removes the "in either order" ambiguity for B_out and B_in: - B_out = G[V_{C'} ∩ L_d]: the boundary of R_{C'} closer to S. - B_in = G[V_{C'} ∩ L_{d+1}]: the boundary farther from S. The outerplanarity step now cites Lemma 2.6 of [bauerfeld-pds] directly (no embedding switch). The "tire structure" step pins the orientation by S's position on the outer face. Remark 1.9 (degenerate cases) updated: the orientation ambiguity is gone, so we state the d=0 case has degenerate B_out and the d=D_max case has degenerate B_in. (R1) and (R2) remain — they are graph-theoretic and unaffected by embedding choice (for 3-connected planar graphs the embedding is essentially unique by Whitney's theorem, so changing the outer face cannot untangle pinches or merge multi-hole topology). 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b/papers/coloring_nested_tire_graphs/paper.tex @@ -174,8 +174,10 @@ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. \begin{lemma}[Tire-component lemma] \label{lem:tire-component} -Let $G$ be a maximal planar graph with fixed embedding $\Pi_G$ and let -$S \subseteq V(G)$ be a level source. For $d \geq 0$, let +Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level +source. Fix a plane embedding $\Pi_G$ of $G$ in which $S$ lies on the +outer face (such an embedding exists for any planar graph and any +single-vertex source). For $d \geq 0$, let \[ G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr] \] @@ -195,21 +197,20 @@ Assume: \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 the closed boundary region are exactly the faces of $G$ in -$F_{C'}$. +Definition~\ref{def:tire-graph}. Its outer boundary +$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$, +namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$; its inner boundary +$B_{\mathrm{in}}$ is the side farther from $S$, namely the +level-$(d+1)$ subgraph $G[V_{C'} \cap L_{d+1}]$; and the triangular +faces of $C$ inside the closed boundary region are exactly the faces of +$G$ in $F_{C'}$. \end{lemma} \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 +\emph{Outerplanarity of the two level parts.} By construction $S$ +lies on the outer face of $\Pi_G$, so Lemma~2.6 of \cite{bauerfeld-pds} +applies directly with $(G, \Pi_G, S)$, giving that $G[L_{d'}]$ is +outerplanar for each $d' \geq 0$. Subgraphs of outerplanar graphs are outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are both outerplanar. @@ -265,21 +266,18 @@ 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). +\emph{Tire structure.} Because $S$ lies on the outer face of $\Pi_G$, +the level-$d$ vertices are closer to $S$ in $\Pi_G$ than the +level-$(d+1)$ vertices; in either the annulus or disk case the +boundary cycle on the $L_d$ side is the boundary of $R_{C'}$ facing +$S$ (the ``outer'' boundary), and the $L_{d+1}$ side is the boundary +facing the interior (the ``inner'' boundary). This identifies +$B_{\mathrm{out}} = G[V_{C'} \cap L_d]$ and $B_{\mathrm{in}} = +G[V_{C'} \cap L_{d+1}]$. In the disk case ($n = 1$) one of the two +level sets is a single vertex (the BFS endpoint at $d = 0$ with +$S = \{v_0\}$, or symmetrically at $d = D_{\max}$ where the inner +side collapses to a deepest vertex), giving the degenerate-boundary +case of Definition~\ref{def:tire-graph}. 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 @@ -291,13 +289,13 @@ $V_{C'} \cap L_{d+1}$ that bound a face of $F_{C'}$. \begin{remark} \label{rem:tire-component-degenerate} Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be -degenerate. For instance, at $d = 0$ with single-vertex source -$S = \{v_0\}$ the unique component of $G'_0$ has -$V_{C'} \cap L_0 = \{v_0\}$ --- the degenerate boundary --- and -$V_{C'} \cap L_1$ a cycle (the link of $v_0$ in $G$). Which of the two -parts is $B_{\mathrm{out}}$ and which is $B_{\mathrm{in}}$ depends on -the orientation of the inherited embedding (equivalently, on which side -of $C$ contains the rest of $\Pi_G$). +degenerate. At $d = 0$ with single-vertex source $S = \{v_0\}$ the +unique component of $G'_0$ has $V_{C'} \cap L_0 = \{v_0\}$ as the +degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the +link of $v_0$ in $G$) as the inner boundary. Symmetrically, at +$d = D_{\max}$, $V_{C'} \cap L_{D_{\max}+1} = \emptyset$ degenerates +to a single deepest vertex serving as the \emph{inner} boundary, with +the level-$D_{\max}$ cycle as the outer boundary. \end{remark} \begin{remark}