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>

papers/coloring_nested_tire_graphs/paper.aux

@@ -25,6 +25,7 @@
 \newlabel{fig:inner-dual-annulus-case}{{4}{9}}
 \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}}
 \newlabel{rem:bridge-case-theta}{{1.14}{9}}
+\newlabel{thm:tread-tree}{{1.15}{10}}
 \bibcite{bauerfeld-depth}{1}
 \bibcite{bauerfeld-nested-tire-duals}{2}
 \newlabel{tocindent-1}{0pt}
@@ -32,5 +33,7 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent }
+\newlabel{rem:tree-multiple-children}{{1.16}{11}}
-\gdef \@abspage@last{10}
+\newlabel{rem:tree-coloring-factorisation}{{1.17}{11}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent }
+\gdef \@abspage@last{11}

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)  27 MAY 2026 02:24
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  27 MAY 2026 02:40
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -511,43 +511,47 @@ Package pdftex.def Info: fig_tire_example.png  used on input line 179.
 
 LaTeX Warning: `h' float specifier changed to `ht'.
 
-[7] [8] [9] [10] (./paper.aux) ) 
+[7] [8] [9]
+Underfull \vbox (badness 10000) has occurred while \output is active []
+
+ [10]
+[11] (./paper.aux) ) 
 Here is how much of TeX's memory you used:
- 14040 strings out of 478268
+ 14043 strings out of 478268
- 279072 string characters out of 5846347
+ 279149 string characters out of 5846347
- 563777 words of memory out of 5000000
+ 563813 words of memory out of 5000000
- 31865 multiletter control sequences out of 15000+600000
+ 31868 multiletter control sequences out of 15000+600000
  477909 words of font info for 61 fonts, out of 8000000 for 9000
  1302 hyphenation exceptions out of 8191
  84i,12n,89p,1156b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s
-</usr/local/texlive/2022/texmf-dist/fonts/type1
+</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
-/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
+onts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
-public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
+nts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
-public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
+nts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
-ublic/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
+ts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
-blic/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
+s/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
-ic/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
+cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
-/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
+/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
-msfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
+mmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmm
-fonts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
+i9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10
-nts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
+.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pf
-s/cm/cmr5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
+b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb><
-m/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
+/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></us
-mr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8
+r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/l
-.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pf
+ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb></usr/loca
-b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
+l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local
-></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb><
+/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/t
-/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></u
+exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/tex
-sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr
+live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texli
-/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr/l
+ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr/local/texlive
-ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/lo
+/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/
-cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/loca
+2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/20
-l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/
+22/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/texliv
-local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
+e/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
-Output written on paper.pdf (10 pages, 594243 bytes).
+Output written on paper.pdf (11 pages, 603044 bytes).
 PDF statistics:
- 165 PDF objects out of 1000 (max. 8388607)
+ 169 PDF objects out of 1000 (max. 8388607)
- 100 compressed objects within 1 object stream
+ 102 compressed objects within 2 object streams
  0 named destinations out of 1000 (max. 500000)
  23 words of extra memory for PDF output out of 10000 (max. 10000000)
 

papers/coloring_nested_tire_graphs/paper.tex

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