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papers: move tire-component lemma + tread partition theorem to foundational paper

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Moved from coloring_nested_tire_dual_graphs/ TO coloring_nested_tire_graphs/:
  - Proposition (Source-side simple-cycle property) → now 1.7
  - Lemma (Tire-component lemma) → now 1.8
  - Theorem (Tire treads partition the bounded faces) → now 1.9
  - Remark (boundaries-may-be-degenerate) → now 1.10
  - Remark (no extra hypotheses needed) → now 1.11

These are foundational structural results about tire-graph
decompositions induced by a level source, not specifically about
the partial tire dual D(T) or coloring.  Belongs in the
foundational paper.

Updates:
  - Internal \cite[Definition~1.5]{bauerfeld-nested-tires} inside
    the moved blocks → local \ref{def:tire-graph}.
  - Foundational paper abstract rewritten to highlight the
    tire-component lemma and tread partition as the main results.
  - Dual paper abstract trimmed: no longer claims the tire-component
    lemma as its own contribution.
  - Dual paper intro citation list adds bullets for the moved
    lemma (\cite[Lemma~1.8]) and theorem (\cite[Theorem~1.9]).
  - No external references to the moved items inside the dual paper.

Page counts:
  - Foundational: 3 → 7 pages.
  - Dual: 9 → 7 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 01:13:25 -04:00
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commit 9185d626af
8 changed files with 425 additions and 406 deletions
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papers/coloring_nested_tire_dual_graphs/paper.aux
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\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\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 tire tread (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}}
\citation{bauerfeld-nested-tires}
\newlabel{def:partial-tire-dual}{{1.1}{2}}
\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}}
\citation{bauerfeld-nested-tires}
\@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{thm:tread-partition}{{1.5}{6}}
\newlabel{rem:tire-component-degenerate}{{1.6}{6}}
\newlabel{prop:edge-vertex-bijection}{{1.3}{2}}
\@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 tire tread (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 {}.}}{3}{}\protected@file@percent }
\newlabel{fig:partial-tire-dual-example}{{1}{3}}
\newlabel{rem:edge-vertex-corollary}{{1.4}{3}}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\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}}
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\@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}}
\@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).}}{4}{}\protected@file@percent }
\newlabel{fig:partial-tire-dual-bridge}{{2}{4}}
\newlabel{def:tire-annular-subgraph}{{1.5}{4}}
\newlabel{def:tire-annular-face-connector}{{1.6}{4}}
\newlabel{def:spokes}{{1.7}{5}}
\newlabel{rem:facial-dual-spoke-only}{{1.8}{5}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{5}{}\protected@file@percent }
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\@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 $u_4, u_5$).}}{6}{}\protected@file@percent }
\newlabel{fig:facial-dual-choices}{{3}{6}}
\newlabel{conj:latin}{{2.1}{6}}
\newlabel{conj:chain-latin}{{2.2}{6}}
\bibcite{bauerfeld-depth}{1}
\bibcite{bauerfeld-nested-tires}{2}
\newlabel{tocindent-1}{0pt}
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papers/coloring_nested_tire_dual_graphs/paper.tex
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\begin{abstract}
This is a follow-up to \cite{bauerfeld-nested-tires}, which
establishes the basic vocabulary of tire graphs $T$ and dual
depth. Building on those definitions, we define the
\emph{partial tire dual} $D(T)$ and analyse its structure in the
spoke-only case (a corona graph $C_{n+m} \circ K_1$), prove the
tire-component lemma
exhibiting every BFS-level component as a tire graph, give an
edge-vertex coloring bijection that reduces counting proper
$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings
of a cycle, and develop the tire-annular-subgraph, face-connector,
and inner/outer-spoke structures in $G'$. A concluding section
records a Latin-substructure conjecture for chain-pigeonhole
compatibility of adjacent tires.
depth, along with the tire-component lemma and the tire-tread
partition theorem. Building on those structural results, we
define the \emph{partial tire dual} $D(T)$ and analyse its
structure in the spoke-only case (a corona graph $C_{n+m} \circ
K_1$), give an edge-vertex coloring bijection that reduces
counting proper $3$-edge-colorings of $D(T)$ to counting proper
$3$-vertex-colorings of a cycle, and develop the
tire-annular-subgraph, face-connector, and inner/outer-spoke
structures in $G'$. A concluding section records a
Latin-substructure conjecture for chain-pigeonhole compatibility
of adjacent tires.
\end{abstract}
\maketitle
@@ -87,7 +87,15 @@ particular we use, without restating, the notions of:
with outer/inner boundaries and annular edges
(\cite[Definition~1.5]{bauerfeld-nested-tires});
\item face/edge counts
(\cite[Remark~1.6]{bauerfeld-nested-tires}).
(\cite[Remark~1.6]{bauerfeld-nested-tires});
\item the \emph{tire-component lemma}
(\cite[Lemma~1.8]{bauerfeld-nested-tires}), which exhibits
each connected component of $G'_d$ as a tire graph
whose tire tread is the union of its depth-$d$ faces;
\item the \emph{tire-tread partition theorem}
(\cite[Theorem~1.9]{bauerfeld-nested-tires}), which shows
the tire treads from a level source partition the bounded
faces of $G$.
\end{itemize}
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
@@ -209,277 +217,6 @@ cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n +
and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof}
\begin{proposition}[Source-side simple-cycle property]
\label{prop:no-level-d-pinch}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let
$C'$ be a connected component of $G'_d$ such that $v$ is incident to
some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident
to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$.
Equivalently: for any such component, the source-side boundary of
$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$).
\end{proposition}
\begin{proof}
Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$
lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices
in $G$ differ in level by at most $1$, so a face at $v$ has depth
exactly $d$ iff both other vertices have level $\geq d$, and depth
$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between
the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of
$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick
$p$ in one such gap and $q$ in another.
The BFS ball $G[L_{<d}]$ is connected, so there exists a simple path
$P$ in $G[L_{<d}]$ from $p$ to $q$. Define the closed walk
\[
W \;:=\; v \to p \to P \to q \to v.
\]
Every vertex of $P$ lies in $L_{<d}$, while $\ell(v) = d$, so $v$ is
distinct from every vertex of $P$; $P$ is simple, so its internal
vertices are distinct; and $p \neq q$ since they lie in different gaps.
Hence $W$ is a simple cycle in $G$.
By the Jordan curve theorem, the planar embedding of $W$ divides $\Pi_G$
into two regions. In $v$'s rotation, the edges $v-p$ and $v-q$ lie at
two specific positions, and they split the rotation into two arcs;
each arc lies in one of the two regions determined by $W$. By choice
of $p, q$, the two arcs of depth-$d$ faces at $v$ in $F_{C'}$ lie in
different regions of $W$ (i.e., one arc on each side).
Since $C'$ is connected in $G'$ and contains depth-$d$ faces in both
arcs, there is a dual path $f_1, f_2, \ldots, f_k$ in $G'_d$ with
$f_1, f_k \in F_{C'}$ incident to $v$ in different arcs, and with the
intermediate faces $f_2, \ldots, f_{k-1}$ not incident to $v$ (a
shortest such dual path). Consecutive faces $f_i, f_{i+1}$ share an
edge $e_i$ of $G$; for $i \geq 2$, both endpoints of $e_i$ lie in
$L_{\geq d}$ (since neither $f_i$ nor $f_{i+1}$ is incident to $v$,
all six vertices of these two triangles lie in $L_{\geq d}$). In
particular, $e_i$ shares no endpoint with $W$ except possibly $v$ ---
and $v$ is excluded from $f_2, \ldots, f_{k-1}$.
A planar edge with neither endpoint on a simple closed planar curve
$W$ has both of its incident faces on the same side of $W$. Applying
this to each $e_i$ ($i \geq 2$) inductively: starting from $f_2$ on
the same side of $W$ as $f_1$ (their shared edge $e_1 = w-w'$ opposite
to $v$ in $f_1$ has $w, w' \in L_{\geq d}$ and hence is not on $W$),
the path $f_2 \to f_3 \to \cdots \to f_{k-1} \to f_k$ stays on one
side of $W$.
But $f_1$ and $f_k$ lie on different sides of $W$ (by construction),
contradicting the conclusion that the entire path lies on one side.
\end{proof}
\begin{lemma}[Tire-component lemma]
\label{lem:tire-component}
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
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\]
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 let $C := G[V_{C'}]$ inherit
its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of
\cite[Definition~1.5]{bauerfeld-nested-tires}. Its outer boundary
$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
boundary closed walk of $O$ in the inherited embedding (a simple cycle
when $O$ is $2$-connected, a non-simple closed walk in general). 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.} By construction $S$
lies on the outer face of $\Pi_G$, so the outerplanarity lemma of
\cite{bauerfeld-depth} 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.
\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
$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
\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.
\emph{Boundary structure.} Each connected component of
$\partial R_{C'}$ traces a closed walk in $G$ that, by the
monochromaticity above, lies entirely in $L_d$ or entirely in
$L_{d+1}$. By Proposition~\ref{prop:no-level-d-pinch}, the depth-$d$
faces of $F_{C'}$ at any $v \in L_d \cap V_{C'}$ form a single
contiguous arc in $v$'s rotation, so the source-side boundary walk
visits each $L_d$-vertex of $V_{C'}$ exactly once: it is a simple
cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
may split into multiple arcs of $v$'s rotation; this corresponds
exactly to $v$ being a cut-vertex of $O$, and the inner-side
boundary walk visits $v$ correspondingly many times --- which is
already accommodated by \cite[Definition~1.5]{bauerfeld-nested-tires} (where
$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
necessarily a simple cycle).
\emph{Outer boundary.} Because $S$ lies on the outer face of $\Pi_G$,
the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$
in the embedding. In the inherited embedding of $C$, the unique
unbounded face is the merged region containing the rest of $\Pi_G$
outside $R_{C'}$ on the $S$ side, so its boundary --- a simple cycle
on $L_d$ (or a single vertex when $V_{C'} \cap L_d = \{v_0\}$, the
$d = 0$ case) --- serves as $B_{\mathrm{out}}$. We set
$B_{\mathrm{out}} := G[V_{C'} \cap L_d]$ if this is a cycle, and
the single vertex $\{v_0\}$ in the degenerate case.
\emph{Inner outerplanar graph.} By the outerplanarity lemma of
\cite{bauerfeld-depth}, $G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O :=
G[V_{C'} \cap L_{d+1}]$. The boundary curve(s) of $R_{C'}$ on the
$L_{d+1}$ side are exactly the boundary of $O$'s outer face in the
inherited embedding; this outer-face boundary is a single closed walk
that traces around $O$ from the outside, traversing any bridge edge
twice and visiting cut-vertices multiple times. This walk is the
inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s
internal structure is needed: when $R_{C'}$ has more than two
boundary components in the surface-classification sense (i.e.\
several disjoint simple cycles on $L_{d+1}$), these correspond
precisely to the multiple connected components or bridge crossings of
$O$, and the outer-face boundary closed walk of $O$ captures them
collectively.
\emph{Tire structure.} The triangular faces of $C$ inside the closed
boundary region 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{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
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}
\label{rem:tire-no-extra-hypotheses}
Two structural features of $R_{C'}$ that might at first appear to
obstruct the tire-graph conclusion are both already accommodated by
\cite[Definition~1.5]{bauerfeld-nested-tires}:
\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may
have the faces of $F_{C'}$ incident to it split into two or more
arcs in $v$'s rotation in $\Pi_G$, separated by faces of higher
depth. This corresponds exactly to $v$ being a cut-vertex of
$O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk
$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each
arc. No additional hypothesis is needed.
\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses
several disjoint depth-$>d$ sub-regions, the inner outerplanar graph
$O$ captures the multi-hole structure as a disconnected or
non-$2$-connected outerplanar graph (with bridges or multiple
components), and its outer-face boundary closed walk serves as
$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices
multiple times.
In the special case $d = 0$ with single-vertex source $S = \{v_0\}$,
$R_{C'}$ is the star of $v_0$, a topological closed disk with one
boundary cycle (the link of $v_0$); the corresponding tire graph has
degenerate outer boundary $\{v_0\}$.
\end{remark}
\begin{proposition}[Edge--vertex coloring bijection for $D(T)$]
\label{prop:edge-vertex-bijection}
papers/coloring_nested_tire_graphs/paper.aux
+12 -5
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@@ -6,6 +6,16 @@
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\citation{bauerfeld-depth}
\citation{bauerfeld-depth}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}}
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\newlabel{tocindent-1}{0pt}
@@ -13,8 +23,5 @@
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\newlabel{rem:tire-counts}{{1.6}{3}}
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@@ -44,15 +44,20 @@
\dedicatory{}
\begin{abstract}
We establish the foundational definitions for studying the
Four Colour Theorem through nested level-structures on plane
triangulations. A \emph{level source} of a triangulation $G$
induces a BFS layering of $G$, which in turn endows the inner
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 the \emph{tire tread} $R$,
a closed region triangulated by the \emph{annular edges}
$E_{\mathrm{ann}}$ --- and record its face/edge counts.
We establish the foundational structure of nested
level-induced tire decompositions of a plane triangulation $G$.
A \emph{level source} of $G$ induces a BFS layering of $G$ and
endows the inner planar dual $G'$ with a \emph{dual depth}
grading. The basic object of study is the \emph{tire graph}
$T$ --- a plane graph whose outer and inner boundaries bound a
closed planar region, the \emph{tire tread} $R$, triangulated by
the \emph{annular edges} $E_{\mathrm{ann}}$. Our main structural
result, the \emph{tire-component lemma}, exhibits each connected
component of $G'_d$ as a tire graph; the \emph{tire-tread
partition theorem} consequence shows the resulting tire treads
partition the bounded faces of $G$. Coloring questions on
$G$ thereby factor through coloring questions on the
individual treads.
\end{abstract}
\maketitle
@@ -190,6 +195,277 @@ 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{proposition}[Source-side simple-cycle property]
\label{prop:no-level-d-pinch}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let
$C'$ be a connected component of $G'_d$ such that $v$ is incident to
some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident
to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$.
Equivalently: for any such component, the source-side boundary of
$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$).
\end{proposition}
\begin{proof}
Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$
lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices
in $G$ differ in level by at most $1$, so a face at $v$ has depth
exactly $d$ iff both other vertices have level $\geq d$, and depth
$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between
the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of
$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick
$p$ in one such gap and $q$ in another.
The BFS ball $G[L_{<d}]$ is connected, so there exists a simple path
$P$ in $G[L_{<d}]$ from $p$ to $q$. Define the closed walk
\[
W \;:=\; v \to p \to P \to q \to v.
\]
Every vertex of $P$ lies in $L_{<d}$, while $\ell(v) = d$, so $v$ is
distinct from every vertex of $P$; $P$ is simple, so its internal
vertices are distinct; and $p \neq q$ since they lie in different gaps.
Hence $W$ is a simple cycle in $G$.
By the Jordan curve theorem, the planar embedding of $W$ divides $\Pi_G$
into two regions. In $v$'s rotation, the edges $v-p$ and $v-q$ lie at
two specific positions, and they split the rotation into two arcs;
each arc lies in one of the two regions determined by $W$. By choice
of $p, q$, the two arcs of depth-$d$ faces at $v$ in $F_{C'}$ lie in
different regions of $W$ (i.e., one arc on each side).
Since $C'$ is connected in $G'$ and contains depth-$d$ faces in both
arcs, there is a dual path $f_1, f_2, \ldots, f_k$ in $G'_d$ with
$f_1, f_k \in F_{C'}$ incident to $v$ in different arcs, and with the
intermediate faces $f_2, \ldots, f_{k-1}$ not incident to $v$ (a
shortest such dual path). Consecutive faces $f_i, f_{i+1}$ share an
edge $e_i$ of $G$; for $i \geq 2$, both endpoints of $e_i$ lie in
$L_{\geq d}$ (since neither $f_i$ nor $f_{i+1}$ is incident to $v$,
all six vertices of these two triangles lie in $L_{\geq d}$). In
particular, $e_i$ shares no endpoint with $W$ except possibly $v$ ---
and $v$ is excluded from $f_2, \ldots, f_{k-1}$.
A planar edge with neither endpoint on a simple closed planar curve
$W$ has both of its incident faces on the same side of $W$. Applying
this to each $e_i$ ($i \geq 2$) inductively: starting from $f_2$ on
the same side of $W$ as $f_1$ (their shared edge $e_1 = w-w'$ opposite
to $v$ in $f_1$ has $w, w' \in L_{\geq d}$ and hence is not on $W$),
the path $f_2 \to f_3 \to \cdots \to f_{k-1} \to f_k$ stays on one
side of $W$.
But $f_1$ and $f_k$ lie on different sides of $W$ (by construction),
contradicting the conclusion that the entire path lies on one side.
\end{proof}
\begin{lemma}[Tire-component lemma]
\label{lem:tire-component}
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
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\]
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 let $C := G[V_{C'}]$ inherit
its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of 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 $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
boundary closed walk of $O$ in the inherited embedding (a simple cycle
when $O$ is $2$-connected, a non-simple closed walk in general). 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.} By construction $S$
lies on the outer face of $\Pi_G$, so the outerplanarity lemma of
\cite{bauerfeld-depth} 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.
\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
$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
\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.
\emph{Boundary structure.} Each connected component of
$\partial R_{C'}$ traces a closed walk in $G$ that, by the
monochromaticity above, lies entirely in $L_d$ or entirely in
$L_{d+1}$. By Proposition~\ref{prop:no-level-d-pinch}, the depth-$d$
faces of $F_{C'}$ at any $v \in L_d \cap V_{C'}$ form a single
contiguous arc in $v$'s rotation, so the source-side boundary walk
visits each $L_d$-vertex of $V_{C'}$ exactly once: it is a simple
cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
may split into multiple arcs of $v$'s rotation; this corresponds
exactly to $v$ being a cut-vertex of $O$, and the inner-side
boundary walk visits $v$ correspondingly many times --- which is
already accommodated by Definition~\ref{def:tire-graph} (where
$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
necessarily a simple cycle).
\emph{Outer boundary.} Because $S$ lies on the outer face of $\Pi_G$,
the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$
in the embedding. In the inherited embedding of $C$, the unique
unbounded face is the merged region containing the rest of $\Pi_G$
outside $R_{C'}$ on the $S$ side, so its boundary --- a simple cycle
on $L_d$ (or a single vertex when $V_{C'} \cap L_d = \{v_0\}$, the
$d = 0$ case) --- serves as $B_{\mathrm{out}}$. We set
$B_{\mathrm{out}} := G[V_{C'} \cap L_d]$ if this is a cycle, and
the single vertex $\{v_0\}$ in the degenerate case.
\emph{Inner outerplanar graph.} By the outerplanarity lemma of
\cite{bauerfeld-depth}, $G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O :=
G[V_{C'} \cap L_{d+1}]$. The boundary curve(s) of $R_{C'}$ on the
$L_{d+1}$ side are exactly the boundary of $O$'s outer face in the
inherited embedding; this outer-face boundary is a single closed walk
that traces around $O$ from the outside, traversing any bridge edge
twice and visiting cut-vertices multiple times. This walk is the
inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s
internal structure is needed: when $R_{C'}$ has more than two
boundary components in the surface-classification sense (i.e.\
several disjoint simple cycles on $L_{d+1}$), these correspond
precisely to the multiple connected components or bridge crossings of
$O$, and the outer-face boundary closed walk of $O$ captures them
collectively.
\emph{Tire structure.} The triangular faces of $C$ inside the closed
boundary region 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{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
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}
\label{rem:tire-no-extra-hypotheses}
Two structural features of $R_{C'}$ that might at first appear to
obstruct the tire-graph conclusion are both already accommodated by
Definition~\ref{def:tire-graph}:
\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may
have the faces of $F_{C'}$ incident to it split into two or more
arcs in $v$'s rotation in $\Pi_G$, separated by faces of higher
depth. This corresponds exactly to $v$ being a cut-vertex of
$O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk
$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each
arc. No additional hypothesis is needed.
\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses
several disjoint depth-$>d$ sub-regions, the inner outerplanar graph
$O$ captures the multi-hole structure as a disconnected or
non-$2$-connected outerplanar graph (with bridges or multiple
components), and its outer-face boundary closed walk serves as
$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices
multiple times.
In the special case $d = 0$ with single-vertex source $S = \{v_0\}$,
$R_{C'}$ is the star of $v_0$, a topological closed disk with one
boundary cycle (the link of $v_0$); the corresponding tire graph has
degenerate outer boundary $\{v_0\}$.
\end{remark}
\begin{thebibliography}{9}