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coloring_nested_tire_graphs: firm up Lemma 1.7 with explicit manifold/boundary hypotheses

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Adds two explicit hypotheses (R1) and (R2) to Lemma 1.7 (tire-component
lemma) and tightens the proof to use them precisely:

(R1) R_{C'} is a topological 2-manifold with boundary; equivalently,
     at every v ∈ V_{C'} the depth-d faces of F_{C'} incident to v form
     a single contiguous arc in v's rotation in Π_G.

(R2) R_{C'} has at most two boundary components.

These rule out, respectively, pinch points (where C' wraps around a
vertex via a global path of depth-d faces, producing a non-manifold
region) and multi-hole topology (a "pair of pants" or worse, which can
occur when several disjoint depth->d lobes sit inside one depth-d
component).

The proof is reorganised into labelled steps:
 1. Outerplanarity of the two level parts (via Lemma 2.6 of [1]).
 2. Layer containment V_{C'} ⊆ L_d ∪ L_{d+1}.
 3. Boundary edges are monochromatic in level (full case analysis
    on the third vertex of the outside face f', using the
    bounded-step property of δ).
 4. Boundary components are simple cycles (uses R1: locally at any
    boundary point R_{C'} is a half-disk, so the boundary walk
    visits each vertex once).
 5. Topological type: planarity + R1 + R2 forces disk or annulus
    via the classification of compact orientable 2-manifolds with
    boundary.
 6. Tire structure: identifies the two boundary parts as the level-d
    and level-(d+1) induced subgraphs, in either order.

Adds Remark 1.10 documenting when (R1) and (R2) hold or fail:
 - (R1) fails iff there is a pinch vertex whose cyclic level sequence
   around it enters and leaves {d, d+1} more than once.
 - (R2) fails iff R_{C'} encloses two or more disjoint depth->d
   sub-regions (the depth-<d region is always a single connected
   BFS ball, so the multi-hole obstruction is always on the away side).
 - Notes the d=0 single-vertex-source case satisfies both hypotheses
   automatically (R_{C'} is the star of v_0).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-25 15:31:59 -04:00
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papers/coloring_nested_tire_graphs/paper.aux
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -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 be the inner-dual subgraph on dual vertices of dual depth $d$, and let
$C'$ be a connected component of $G'_d$. Write $C'$ be a connected component of $G'_d$. Write
$F_{C'} := \{f : d_f \in V(C')\}$, $F_{C'} := \{f : d_f \in V(C')\}$,
$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and $V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit
$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$. 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 Assume:
sense of Definition~\ref{def:tire-graph}: the two boundary parts \begin{itemize}
$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$ \item[\emph{(R1)}] $R_{C'}$ is a topological $2$-manifold with boundary;
subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph 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 $G[V_{C'} \cap L_{d+1}]$, 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
$F_{C'}$. $F_{C'}$.
\end{lemma} \end{lemma}
\begin{proof}[Proof sketch] \begin{proof}
Since $S$ is a single vertex, we may choose a plane embedding of $G$ \emph{Outerplanarity of the two level parts.} Since $S$ is a single
that places $S$ on the outer face. By Lemma~2.6 of vertex, choose a plane embedding of $G$ with $S$ on the outer face.
\cite{bauerfeld-pds}, applied with this embedding and source set $S$, By Lemma~2.6 of \cite{bauerfeld-pds} applied with this embedding and
the subgraph $G[L_{d'}]$ is outerplanar for each $d' \geq 0$; source $S$, $G[L_{d'}]$ is outerplanar for each $d' \geq 0$;
outerplanarity is a graph property, so this conclusion is independent outerplanarity is a graph property, so the conclusion is independent
of the embedding choice. Since subgraphs of outerplanar graphs are of the embedding choice. Subgraphs of outerplanar graphs are
outerplanar, both $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
outerplanar. both outerplanar.
Layer containment (a consequence of the bounded-step property of BFS \emph{Layer containment.} Each $f \in F_{C'}$ has at least one vertex
on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$ at level $d$, and adjacent vertices in $G$ differ in level by at most
has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap $1$; combined with $\delta_G(d_f) = d$, this forces
L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least $V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup
one vertex in $L_d$. 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 \emph{Boundary edges are monochromatic in level.} Each edge $e$ on
F_{C'}} f \subseteq |\Pi_G|$. Each edge on the topological boundary $\partial R_{C'}$ separates a face $f \in F_{C'}$ from a face
$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a $f' \notin F_{C'}$. Because $f$ and $f'$ share the edge $e$, their
face $f' \notin F_{C'}$. Such an $f'$ has dual depth in $\{d-1, d+1\}$: dual vertices are adjacent in $G'$; if both had depth $d$ they would
if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a lie in the same component of $G'_d$, contradicting $d_{f} \in C'$ and
distinct component of $G'_d$, which contradicts the connectivity of $d_{f'} \notin C'$. Hence $\delta_G(d_{f'}) \neq d$; combined with
$C'$ together with the fact that adjacent depth-$d$ dual vertices belong the bounded-step property of $\delta$ across $G'$-adjacent faces,
to the same component. A short case analysis on the level of the third $\delta_G(d_{f'}) \in \{d-1, d+1\}$.
vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have \begin{itemize}
both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both \item If $\delta_G(d_{f'}) = d - 1$, the third vertex $w$ of
endpoints in $L_{d+1}$. Each connected component of $\partial R_{C'}$ $f' = \{u, v, w\}$ (where $u, v$ are the endpoints of $e$) has
is therefore monochromatic in level. $\ell(w) = d - 1$. 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 \emph{Boundary components are simple cycles.} By hypothesis (R1),
of triangles glued along edges, $\partial R_{C'}$ consists of either $R_{C'}$ is a $2$-manifold with boundary, so locally at any boundary
one closed walk (when $R_{C'}$ is a topological disk) or two closed point $p$ the region $R_{C'}$ is homeomorphic to a half-disk and the
walks (when $R_{C'}$ is a topological annulus). These walks are link of $p$ in $\partial R_{C'}$ is an arc with two endpoints. In
simple cycles in $G$ on the respective level sets (with one cycle particular, at every boundary vertex $v$ exactly two boundary edges
possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the are incident, and the boundary walk traverses $v$ exactly once. Each
endpoints of the BFS, giving the degenerate-boundary case of boundary component is therefore a simple closed walk in $G$ --- a
Definition~\ref{def:tire-graph}). Together with the outerplanarity of simple cycle, possibly degenerating to a single vertex if $v$ has no
$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above, incident boundary edges (which happens precisely at the BFS endpoints
these cycles serve as the two boundary parts of $C$, in either order, $d = 0$ with $S = \{v_0\}$, or where an entire level set $V_{C'} \cap
and the depth-$d$ triangles in $F_{C'}$ tile the closed region between L_{d+1}$ is empty).
them.
\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} \end{proof}
\begin{remark} \begin{remark}
@@ -249,6 +300,42 @@ the orientation of the inherited embedding (equivalently, on which side
of $C$ contains the rest of $\Pi_G$). of $C$ contains the rest of $\Pi_G$).
\end{remark} \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} \begin{thebibliography}{9}
\bibitem{bauerfeld-pds} \bibitem{bauerfeld-pds}