coloring_nested_tire_graphs: theorem on tire-tread rooted tree structure
NEW Theorem 1.15: The tire treads from a single-vertex level
source S = {v_0} form a rooted tree T(G, S) under face containment.
Statement:
- Root: the depth-0 tire tread T_0 with degenerate outer
boundary {v_0} (the apex tire, B_out = {v_0}).
- Parent: for any tire tread T_c at depth d ≥ 1, the unique
parent T_p at depth d-1 is the tire whose inner outerplanar
graph O^(p) has B_out^(c) as one of its bounded faces.
Equivalently, R_c lies inside this bounded face of O^(p).
- Children: bijection with bounded faces of O^(p) whose
interior contains depth-≥(d+2) vertices.
Proof structure:
1. Root well-defined: G'_0 is connected (fan around v_0), so
unique component → unique T_0.
2. Existence of parent: faces immediately outside B_out^(c) on
the S-side have depth d-1, lie in some component of G'_{d-1}.
3. Uniqueness: by Proposition 1.7 (source-side simple-cycle
property), B_out^(c) is a simple cycle, and the depth-(d-1)
faces around it form a single contiguous arc in the dual,
hence one component → unique parent.
4. Children description: bounded faces of O^(p) are in bijection
with deeper component-tires.
5. Tree property: parent map strictly decreases depth, hence
no cycles, hence rooted tree.
Plus two clarifying remarks:
- Remark 1.16: multiple children iff O^(p) has multiple bounded
faces with non-trivial interiors. Spoke-only case → exactly
one child.
- Remark 1.17: combined with Theorem 1.9 (partition) and
Theorem 1.12 (outerplanar inner dual), any coloring problem
on G factors through:
• local outerplanar coloring on each tread,
• parent-child consistency along shared B_out^(c) cycles.
This is the structural setup for the chain-pigeonhole program.
Page count: 10 → 11.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
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\newlabel{rem:bridge-case-theta}{{1.14}{9}}
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\newlabel{thm:tread-tree}{{1.15}{10}}
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@@ -757,6 +757,122 @@ and so contributes no degree-$2$ branch vertex), hence is
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outerplanar as predicted.
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\end{remark}
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\begin{theorem}[Tire treads form a rooted tree under face containment]
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\label{thm:tread-tree}
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Let $G$ be a maximal planar graph with planar embedding $\Pi_G$
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and let $S \subseteq V(G)$ be a single-vertex level source
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$\{v_0\}$ lying on the outer face of $\Pi_G$. The collection
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$\mathcal{R}(G, S)$ of tire treads
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(Theorem~\ref{thm:tread-partition}) carries a canonical rooted
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tree structure $\mathcal{T}(G, S)$ defined as follows.
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\begin{itemize}
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\item \emph{Root.} The depth-$0$ tire tread $T_0$ --- the unique
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tire produced by Lemma~\ref{lem:tire-component} at $d = 0$,
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with degenerate outer boundary $B_{\mathrm{out}} = \{v_0\}$
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and inner outerplanar graph $O^{(0)} = G[L_1]$ --- is the
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root.
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\item \emph{Parent.} For each tire tread $T_c$ at depth $d \ge 1$,
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its outer boundary $B_{\mathrm{out}}^{(c)}$ is a cycle in
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$L_d$. The \emph{parent} of $T_c$ is the unique tire tread
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$T_p$ at depth $d - 1$ whose inner outerplanar graph
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$O^{(p)}$ has $B_{\mathrm{out}}^{(c)}$ as the boundary cycle
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of one of its bounded faces. Equivalently, $R_c$ lies
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inside this bounded face of $O^{(p)}$ (which is itself the
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region of the plane cut off by $B_{\mathrm{out}}^{(c)}$ on
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the side away from $S$).
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\item \emph{Children.} The children of a tire tread $T_p$ are in
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bijection with those bounded faces of $O^{(p)}$ whose
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interiors contain at least one vertex of $G$ at level
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$\ge d + 2$ --- equivalently, with the connected components
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of $G'_{d+1}$ whose tires have outer boundary cycle equal to
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a bounded face of $O^{(p)}$.
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\end{itemize}
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Every tire tread except $T_0$ has exactly one parent; a tire
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tread may have zero, one, or several children.
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\end{theorem}
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\begin{proof}
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\emph{Root is well-defined.} At $d = 0$ with single-vertex source
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$S = \{v_0\}$, the dual subgraph $G'_0$ is connected (every face
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of $G$ incident to $v_0$ has dual depth $0$, and they form a
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single fan around $v_0$). By
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Lemma~\ref{lem:tire-component}, the unique component of $G'_0$
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gives the depth-$0$ tire $T_0$ described above.
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\emph{Existence of parent.} Fix a tire tread $T_c$ at depth $d
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\ge 1$ arising from a connected component $C'_c$ of $G'_d$. Its
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outer boundary $B_{\mathrm{out}}^{(c)} = G[V_{C'_c} \cap L_d]$ is
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a simple cycle in $L_d$ (Lemma~\ref{lem:tire-component}; the
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source-side boundary of a tire is always a simple cycle, by
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Proposition~\ref{prop:no-level-d-pinch}). The faces of $G$
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immediately outside $B_{\mathrm{out}}^{(c)}$ on the side facing $S$
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have depth $d - 1$ (one of their three vertices lies in $L_{d-1}$,
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two in $L_d$). Let $C'_p$ be the connected component of
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$G'_{d-1}$ containing the dual vertex of any such face.
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\emph{Uniqueness of parent.} $B_{\mathrm{out}}^{(c)}$ is a single
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simple cycle in $G$, with a well-defined ``$S$-side'' (the side
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of the cycle closer to $v_0$ in $\Pi_G$). The depth-$(d-1)$
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faces lying on this side form a single contiguous arc around
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$B_{\mathrm{out}}^{(c)}$ in the dual --- they are all $G'$-adjacent
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in sequence (each pair of consecutive arc faces shares an edge in
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$B_{\mathrm{out}}^{(c)}$). Hence they all lie in the same
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connected component $C'_p$ of $G'_{d-1}$, which is therefore
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unique.
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\emph{$B_{\mathrm{out}}^{(c)}$ bounds a face of $O^{(p)}$.} The
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parent tire $T_p$ has $V(O^{(p)}) = V_{C'_p} \cap L_d \supseteq
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V(B_{\mathrm{out}}^{(c)})$. The cycle $B_{\mathrm{out}}^{(c)}$ is
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a subgraph of $O^{(p)}$ that bounds a face of $O^{(p)}$ in the
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inherited embedding: the cycle traces around a depth-$\ge d+1$
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region (containing $R_c$ and any descendants of $T_c$), which is
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exactly a bounded face of $O^{(p)}$.
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\emph{Children description.} The bounded faces of $O^{(p)}$ are
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in bijection with the connected components of $G'_d$ whose
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faces lie inside those bounded regions (= one component per
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bounded face, by an argument analogous to the existence-and-
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uniqueness step above, applied one level deeper).
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\emph{Tree property.} Every non-root $T_c$ has a unique parent at
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strictly smaller depth. Iterating the parent map strictly
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decreases depth, terminating at $T_0$. No cycles can form
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(depth is monotone). Hence $\mathcal{T}(G, S)$ is a rooted tree.
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\end{proof}
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\begin{remark}
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\label{rem:tree-multiple-children}
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A parent tire $T_p$ has multiple children precisely when its
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inner outerplanar graph $O^{(p)}$ has multiple bounded faces with
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non-trivial interiors (= containing depth-$\ge d+2$ vertices of
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$G$). This happens, for instance, when $O^{(p)}$ has chords or
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cut-vertices that subdivide its inner region, or when $O^{(p)}$
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has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$.
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By contrast, if $O^{(p)}$ is a simple cycle (the spoke-only case
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of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty
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interior, $T_p$ has exactly one child.
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\end{remark}
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\begin{remark}
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\label{rem:tree-coloring-factorisation}
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Combining Theorem~\ref{thm:tread-partition} (treads partition
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the bounded faces of $G$) with
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Theorem~\ref{thm:tread-tree} (treads form a rooted tree), any
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proper coloring problem on $G$'s bounded faces factors through:
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\begin{itemize}
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\item local coloring problems on each tread (the inner dual of
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each tread is outerplanar by
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Theorem~\ref{thm:inner-dual-outerplanar}), plus
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\item consistency constraints along parent-child interfaces (the
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cycle $B_{\mathrm{out}}^{(c)}$ shared between a child and the
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face of its parent's $O^{(p)}$).
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\end{itemize}
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This is the structural setup underlying the chain-pigeonhole
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program for tire treads.
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\end{remark}
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\begin{thebibliography}{9}
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