coloring_nested_tire_graphs: notation cleanup pass

Define previously-implicit objects and unify conventions: - define level sets L_d (and L_{<d}, L_{>=d}) in the Levels definition - factor G'_d, F_{C'}, V_{C'}, R_{C'} into a standalone definition before Prop 1.6, removing the forward reference - name the annular faces F_ann and state the tire-graph tuple form T = (B_out, O, E_ann) in the tire-graph definition - ground the full tire dual D(T) where Gamma is introduced - normalize tree superscripts (0)/(p)/(c) to the tire-symbol form (T_0)/(T_p)/(T_c) - resolve the boundary-count clash: use nu = |V(B_in)| (inner) and mu = |V(B_out)| (outer) throughout, freeing n for |V(G)| Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
2026-05-29 23:38:12 -04:00
parent 454c79b289
commit 92f5efc3f1
4 changed files with 165 additions and 126 deletions
@@ -1,43 +1,43 @@
 \relax 
 \citation{bauerfeld-nested-tire-duals}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
-\newlabel{def:dual}{{1.3}{1}}
+\newlabel{def:dual}{{1.3}{2}}
 \newlabel{def:dual-depth}{{1.4}{2}}
 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname  \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
 \newlabel{fig:dual-depth}{{1}{2}}
-\newlabel{def:tire-graph}{{1.5}{2}}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm  {out}}$ a $6$-cycle on vertices $0,\dots  ,5$ (blue), inner boundary $B_{\mathrm  {in}}$ a $4$-cycle on vertices $6,\dots  ,9$ (red), inner outerplanar graph $O = B_{\mathrm  {in}} \cup \{7\text  {--}9\}$ (with one chord, orange), and $E_{\mathrm  {ann}}$ (grey) tiling the annulus between $B_{\mathrm  {out}}$ and $B_{\mathrm  {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent }
-\newlabel{fig:tire-example}{{2}{3}}
-\newlabel{rem:tire-counts}{{1.6}{3}}
-\newlabel{prop:no-level-d-pinch}{{1.7}{3}}
-\newlabel{lem:tire-component}{{1.8}{4}}
+\newlabel{def:dual-component}{{1.5}{2}}
+\newlabel{def:tire-graph}{{1.6}{3}}
+\newlabel{rem:tire-counts}{{1.7}{3}}
+\newlabel{prop:no-level-d-pinch}{{1.8}{3}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm  {out}}$ a $6$-cycle on vertices $0,\dots  ,5$ (blue), inner boundary $B_{\mathrm  {in}}$ a $4$-cycle on vertices $6,\dots  ,9$ (red), inner outerplanar graph $O = B_{\mathrm  {in}} \cup \{7\text  {--}9\}$ (with one chord, orange), and $E_{\mathrm  {ann}}$ (grey) tiling the annulus between $B_{\mathrm  {out}}$ and $B_{\mathrm  {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent }
+\newlabel{fig:tire-example}{{2}{4}}
 \citation{bauerfeld-depth}
+\newlabel{lem:tire-component}{{1.9}{5}}
 \citation{bauerfeld-depth}
-\newlabel{thm:tread-partition}{{1.9}{6}}
-\newlabel{rem:tire-component-degenerate}{{1.10}{6}}
-\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}}
-\newlabel{thm:inner-dual-outerplanar}{{1.12}{7}}
+\newlabel{thm:tread-partition}{{1.10}{6}}
+\newlabel{rem:tire-component-degenerate}{{1.11}{7}}
+\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
+\newlabel{thm:inner-dual-outerplanar}{{1.13}{7}}
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the centre; the non-degenerate boundary $B_{\mathrm  {non-deg}}$ (red) is the hexagonal outer cycle; spokes (grey) triangulate the disk into a fan of $6$ triangles around $v_0$. Each triangle has two spoke edges (interior, contributing $\Gamma $-edges) and one boundary edge (contributing a leaf in $D(T)$, no $\Gamma $-edge). The inner dual $\Gamma $ (blue) is the cycle $C_6$ formed by the six annular face centroids, a manifestly outerplanar graph.}}{8}{}\protected@file@percent }
 \newlabel{fig:inner-dual-disk-case}{{3}{8}}
 \citation{bauerfeld-nested-tire-duals}
 \citation{bauerfeld-nested-tire-duals}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm  {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text  {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph  {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong  \Theta (1, 7, 7)$ is outerplanar.}}{9}{}\protected@file@percent }
-\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}}
 \citation{tait-original}
-\newlabel{thm:tait-tire}{{1.15}{10}}
-\newlabel{rem:count-general-outerplanar}{{1.16}{10}}
-\newlabel{thm:tread-tree}{{1.17}{10}}
-\newlabel{rem:tree-multiple-children}{{1.18}{11}}
-\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
-\newlabel{rem:tree-coloring-factorisation}{{1.20}{13}}
-\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.19\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal  {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal  {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{14}{}\protected@file@percent }
-\newlabel{fig:tire-tree-decomposition}{{5}{14}}
-\newlabel{def:seam}{{1.21}{14}}
-\newlabel{def:partial-tire-tree}{{1.22}{15}}
-\newlabel{lem:seam-edge-shared}{{1.23}{15}}
-\newlabel{conj:seam-counterexample}{{1.24}{15}}
+\newlabel{rem:hamilton-cycle-spoke-only}{{1.14}{9}}
+\newlabel{rem:bridge-case-theta}{{1.15}{9}}
+\newlabel{thm:tait-tire}{{1.16}{9}}
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm  {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text  {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph  {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong  \Theta (1, 7, 7)$ is outerplanar.}}{10}{}\protected@file@percent }
+\newlabel{fig:inner-dual-annulus-case}{{4}{10}}
+\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
+\newlabel{thm:tread-tree}{{1.18}{11}}
+\newlabel{rem:tree-multiple-children}{{1.19}{12}}
+\newlabel{thm:tire-tree-decomposition}{{1.20}{12}}
+\newlabel{rem:tree-coloring-factorisation}{{1.21}{14}}
+\newlabel{def:seam}{{1.22}{14}}
+\newlabel{def:partial-tire-tree}{{1.23}{14}}
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal  {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal  {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
+\newlabel{fig:tire-tree-decomposition}{{5}{15}}
+\newlabel{lem:seam-edge-shared}{{1.24}{15}}
 \bibcite{tait-original}{1}
 \bibcite{bauerfeld-depth}{2}
 \bibcite{bauerfeld-nested-tire-duals}{3}
@@ -46,5 +46,6 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
+\newlabel{conj:seam-counterexample}{{1.25}{16}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent }
 \gdef \@abspage@last{16}
@@ -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 22:18
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  29 MAY 2026 23:25
 entering extended mode
 restricted \write18 enabled.
 %&-line parsing enabled.
@@ -495,76 +495,88 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
 e
 ))
 [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
-<fig_dual_depth.png, id=25, 642.8015pt x 606.265pt>
+<fig_dual_depth.png, id=26, 642.8015pt x 606.265pt>
 File: fig_dual_depth.png Graphic file (type png)
 <use fig_dual_depth.png>
-Package pdftex.def Info: fig_dual_depth.png  used on input line 128.
+Package pdftex.def Info: fig_dual_depth.png  used on input line 131.
 (pdftex.def)             Requested size: 251.9989pt x 237.67276pt.

 [2 <./fig_dual_depth.png>]
-<fig_tire_example.png, id=31, 559.64081pt x 375.804pt>
+<fig_tire_example.png, id=32, 559.64081pt x 375.804pt>
 File: fig_tire_example.png Graphic file (type png)
 <use fig_tire_example.png>
-Package pdftex.def Info: fig_tire_example.png  used on input line 183.
+Package pdftex.def Info: fig_tire_example.png  used on input line 210.
 (pdftex.def)             Requested size: 280.79956pt x 188.56097pt.
- [3 <./fig_tire_example.png>] [4] [5] [6]
+

 LaTeX Warning: `h' float specifier changed to `ht'.

-[7] [8] [9] [10] [11] [12]
-<fig_tire_tree_decomposition.png, id=72, 1101.3145pt x 633.9685pt>
+[3] [4 <./fig_tire_example.png>] [5] [6]
+
+LaTeX Warning: `h' float specifier changed to `ht'.
+
+[7] [8]
+
+LaTeX Warning: `h' float specifier changed to `ht'.
+
+[9]
+Underfull \vbox (badness 10000) has occurred while \output is active []
+
+ [10]
+[11] [12] [13]
+<fig_tire_tree_decomposition.png, id=76, 1101.3145pt x 633.9685pt>
 File: fig_tire_tree_decomposition.png Graphic file (type png)
 <use fig_tire_tree_decomposition.png>
-Package pdftex.def Info: fig_tire_tree_decomposition.png  used on input line 10
-98.
+Package pdftex.def Info: fig_tire_tree_decomposition.png  used on input line 11
+24.
 (pdftex.def)             Requested size: 341.9989pt x 196.86678pt.


 LaTeX Warning: `h' float specifier changed to `ht'.

-[13] [14 <./fig_tire_tree_decomposition.png>]
-Overfull \hbox (1.78508pt too wide) in paragraph at lines 1228--1230
+[14] [15 <./fig_tire_tree_decomposition.png>]
+Overfull \hbox (1.78508pt too wide) in paragraph at lines 1254--1256
 []\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr
 ivial seam $\OML/cmm/m/it/10 C$ \OT1/cmr/m/it/10 of $\OML/cmm/m/it/10 G$ \OT1/c
 mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m
 /it/10 C\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^U
 []

-[15] [16] (./paper.aux) ) 
+[16] (./paper.aux) ) 
 Here is how much of TeX's memory you used:
- 14061 strings out of 478268
- 279616 string characters out of 5846347
- 563910 words of memory out of 5000000
- 31884 multiletter control sequences out of 15000+600000
+ 14062 strings out of 478268
+ 279636 string characters out of 5846347
+ 565983 words of memory out of 5000000
+ 31885 multiletter control sequences out of 15000+600000
 478218 words of font info for 62 fonts, out of 8000000 for 9000
 1302 hyphenation exceptions out of 8191
 84i,12n,89p,1168b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s
-</usr/local/texlive/2022/texmf-dist/fonts/type1/public
-/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
-amsfonts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
-sfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
-sfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
-fonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
-onts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
-ts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
-/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
-m/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
-cmmi9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
-r10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5
-.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pf
-b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb><
-/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></us
-r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb></usr/l
-ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/lo
-cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/loca
-l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/
-texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/te
-xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr/local/texl
-ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texli
-ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive
-/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/tex
-live/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
-Output written on paper.pdf (16 pages, 927226 bytes).
+</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
+onts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
+nts/cm/cmbx8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
+s/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
+s/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
+/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
+cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
+/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
+mmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmm
+i8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9
+.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.p
+fb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb>
+</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></u
+sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/
+local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/loc
+al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb></usr/local/
+texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/t
+exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/tex
+live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texli
+ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive
+/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr/local/texlive/2
+022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/20
+22/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022
+/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/texlive/
+2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
+Output written on paper.pdf (16 pages, 929392 bytes).
 PDF statistics:
 192 PDF objects out of 1000 (max. 8388607)
 116 compressed objects within 2 object streams
@@ -98,7 +98,10 @@ A \emph{level source} of $G$ is a set $S \subseteq V$ that is either
 \begin{definition}[Levels]
 Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
 $\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
-source vertex.
+source vertex.  We write $L_d := \{v \in V : \ell_G(v) = d\}$ for the
+\emph{level-$d$ vertex set}, and abbreviate $L_{<d} := \bigcup_{d' < d}
+L_{d'}$ and $L_{\geq d} := \bigcup_{d' \geq d} L_{d'}$ (similarly
+$L_{>d}$, $L_{\leq d}$).
 \end{definition}

 \begin{definition}[Dual]
@@ -136,6 +139,23 @@ vertex.}
 \label{fig:dual-depth}
 \end{figure}

+\begin{definition}[Depth-$d$ dual subgraph and its components]
+\label{def:dual-component}
+For $d \geq 0$, the \emph{depth-$d$ dual subgraph} is
+\[
+    G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr],
+\]
+the inner-dual subgraph induced on the dual vertices of dual depth $d$.
+For a connected component $C'$ of $G'_d$ we write
+\[
+    F_{C'} := \{f : d_f \in V(C')\}, \qquad
+    V_{C'} := \bigcup_{f \in F_{C'}} V(f),
+\]
+for its set of faces and the vertices of $G$ bounding them, and
+$R_{C'} := \bigcup_{f \in F_{C'}} f \subseteq |\Pi_G|$ for the closed
+planar region these faces cover.
+\end{definition}
+
 \begin{definition}[Tire graph]
 \label{def:tire-graph}
 A \emph{tire graph} consists of a plane graph $T$ together with an
@@ -164,7 +184,9 @@ where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
 property that, in the plane embedding of $T$, the closed planar
 region $R$ bounded externally by $B_{\mathrm{out}}$ and internally
 by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$
-whose union is $R$.  We call $R$ the \emph{tire tread} of $T$.
+whose union is $R$.  We call $R$ the \emph{tire tread} of $T$ and write
+$F_{\mathrm{ann}}$ for this set of triangular faces (the \emph{annular
+faces}).

 When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected,
 the tread is a closed annulus.  More generally, $R$ is a closed
@@ -176,6 +198,11 @@ definition accommodates outerplanar inner graphs with bridges,
 cut-vertices, or multiple connected components.  When either
 boundary is degenerate, the tread is a closed disk with that vertex
 as apex.
+
+We summarize the data of a tire graph as the triple
+$T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$, from which $B_{\mathrm{in}}$,
+the annular faces $F_{\mathrm{ann}}$, and the tread $R$ are determined;
+we freely identify a tire graph with its underlying plane graph $T$.
 \end{definition}

 \begin{figure}[h]
@@ -193,12 +220,12 @@ triangular faces.}

 \begin{remark}
 \label{rem:tire-counts}
-Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$.  By
-Euler's formula on the tire tread $R$, the tire graph has $m + k$
-triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
+Let $\mu = |V(B_{\mathrm{out}})|$ and $\nu = |V(B_{\mathrm{in}})|$.  By
+Euler's formula on the tire tread $R$, the tire graph has $\mu + \nu$
+triangular faces inside $R$ and $|E_{\mathrm{ann}}| = \mu + \nu$
 annular edges when neither boundary is degenerate; when exactly one
-boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
-triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
+boundary is degenerate (so $\min(\mu, \nu) = 1$), there are $\mu + \nu - 1$
+triangular faces and $|E_{\mathrm{ann}}| = \mu + \nu - 1$.
 \end{remark}

 \begin{proposition}[Source-side simple-cycle property]
@@ -268,16 +295,11 @@ contradicting the conclusion that the entire path lies on one side.
 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 $R_{C'} := \bigcup_{f \in F_{C'}} f
-\subseteq |\Pi_G|$.
+single-vertex source).  For $d \geq 0$, let $C'$ be a connected
+component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$,
+bounding vertices $V_{C'}$, and region $R_{C'}$ as in
+Definition~\ref{def:dual-component}; let $C := G[V_{C'}]$ inherit its
+embedding from $\Pi_G$.

 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 $\Pi_G$,
@@ -478,8 +500,11 @@ Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph,
 and let $\Gamma$ be the graph on vertex set
 $\{d_f : f \in F_{\mathrm{ann}}\}$ with an edge $d_f d_{f'}$ for
 each interior annular edge of $T$ (= each edge of $T$ whose two
-incident faces both lie in $F_{\mathrm{ann}}$).  Then $\Gamma$ is
-outerplanar.
+incident faces both lie in $F_{\mathrm{ann}}$).  Equivalently,
+$\Gamma$ is the subgraph induced on $F_{\mathrm{ann}}$ of the
+\emph{full tire dual} $D(T)$ --- the dual of $T$ taken over all of
+its triangular faces, in which each boundary edge of $R$ contributes
+a degree-$1$ vertex.  Then $\Gamma$ is outerplanar.

 Moreover, $\Gamma$ admits a planar embedding as a (possibly
 non-simple) Hamilton walk through every $d_f$, plus zero or more
@@ -569,12 +594,12 @@ $\partial R = B_{\mathrm{out}} \sqcup \overline{B_{\mathrm{in}}}$
 appropriate orientation).  Each boundary edge of $R$ is incident to
 exactly one annular face: walking around $B_{\mathrm{out}}$ in
 cyclic order produces a sequence
-$f^{\mathrm{out}}_1, f^{\mathrm{out}}_2, \dots, f^{\mathrm{out}}_n$
+$f^{\mathrm{out}}_1, f^{\mathrm{out}}_2, \dots, f^{\mathrm{out}}_\mu$
 of (not necessarily distinct) annular faces, one per
 $B_{\mathrm{out}}$-edge; similarly walking around
 $B_{\mathrm{in}}$ produces a sequence
-$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}$ where
-$m_\partial$ is the length of the inner-boundary walk.  Pick any
+$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial}$ where
+$\nu_\partial$ is the length of the inner-boundary walk.  Pick any
 spoke $e^\star = u w \in E_{\mathrm{ann}}$ with $u \in
 V(B_{\mathrm{out}})$ and $w \in V(B_{\mathrm{in}})$; cut $R$ along
 $e^\star$.  This converts the annulus into a closed disk
@@ -584,8 +609,8 @@ and once back along $e^\star$.  Concatenating the two boundary
 sequences (in the order dictated by this disk traversal) yields a
 single cyclic sequence
 \[
-   \mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_n,
-       f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial})
+   \mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_\mu,
+       f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{\nu_\partial})
 \]
 of annular faces with multiplicities.

@@ -742,7 +767,7 @@ $O$ $2$-connected and $E_{\mathrm{ann}}$ consisting only of spokes),
 every annular face has exactly one boundary edge, every
 $d_f$ has $\Gamma$-degree $2$, and the construction of the
 Theorem~\ref{thm:inner-dual-outerplanar} proof reduces to the
-classical Hamilton cycle $\Gamma \cong C_{n+m}$ with zero chords.
+classical Hamilton cycle $\Gamma \cong C_{\mu+\nu}$ with zero chords.
 \end{remark}

 \begin{remark}
@@ -753,7 +778,7 @@ an interior annular edge in $\Gamma$ rather than two leaves in
 $D(T)$ (see Definition~1.7 of \cite{bauerfeld-nested-tire-duals}).
 The two bridge-incident annular triangles have $\Gamma$-degree $3$;
 the resulting $\Gamma$ has the structure of a Hamilton cycle of
-length $n + m_\partial$ plus a single chord (length $1$).  This
+length $\mu + \nu_\partial$ plus a single chord (length $1$).  This
 corresponds to the theta graph $\Theta(1, b, c)$ identified
 empirically in \cite{bauerfeld-nested-tire-duals}, which has no
 $K_{2,3}$ subdivision (since one of the three paths has length $1$
@@ -823,13 +848,14 @@ stated equality.
 \label{rem:count-general-outerplanar}
 Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a
 tire to the $3$-edge-coloring count of its outerplanar inner dual
-$\Gamma$.  For the cycle case $\Gamma \cong C_n$ (the spoke-only
+$\Gamma$.  For the cycle case $\Gamma \cong C_{\mu+\nu}$ (the spoke-only
 case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle
-chromatic polynomial at $k = 3$ gives
-$2^n + 2 (-1)^n$.  For an inner dual with one or more non-crossing
-chords, the count depends on the chord structure, not just on the
-pair (number of vertices, number of chords): two outerplanar graphs
-with the same $n$ and number of chords can have different proper
+chromatic polynomial at $3$ colours gives
+$2^{\mu+\nu} + 2 (-1)^{\mu+\nu}$.  For an inner dual with one or more
+non-crossing chords, the count depends on the chord structure, not just
+on the pair (number of vertices, number of chords): two outerplanar
+graphs with the same number of vertices and number of chords can have
+different proper
 $3$-edge-coloring counts depending on how the chords are arranged
 (nested, sequential, sharing vertices, etc.).  Every such count
 can nevertheless be computed in linear time by tree-decomposition
@@ -851,23 +877,23 @@ tree structure $\mathcal{T}(G, S)$ defined as follows.
 \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
+      and inner outerplanar graph $O^{(T_0)} = G[L_1]$ --- 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
+      its outer boundary $B_{\mathrm{out}}^{(T_c)}$ 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
+      $O^{(T_p)}$ has $B_{\mathrm{out}}^{(T_c)}$ 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
+      inside this bounded face of $O^{(T_p)}$ (which is itself the
+      region of the plane cut off by $B_{\mathrm{out}}^{(T_c)}$ 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
+      bijection with those bounded faces of $O^{(T_p)}$ 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 $O^{(p)}$.
+      a bounded face of $O^{(T_p)}$.
 \end{itemize}

 Every tire tread except $T_0$ has exactly one parent; a tire
@@ -884,34 +910,34 @@ 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
+outer boundary $B_{\mathrm{out}}^{(T_c)} = G[V_{C'_c} \cap L_d]$ 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 $S$
+immediately outside $B_{\mathrm{out}}^{(T_c)}$ 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
+\emph{Uniqueness of parent.}  $B_{\mathrm{out}}^{(T_c)}$ 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
+$B_{\mathrm{out}}^{(T_c)}$ 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
+$B_{\mathrm{out}}^{(T_c)}$).  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
+\emph{$B_{\mathrm{out}}^{(T_c)}$ bounds a face of $O^{(T_p)}$.}  The
+parent tire $T_p$ has $V(O^{(T_p)}) = V_{C'_p} \cap L_d \supseteq
+V(B_{\mathrm{out}}^{(T_c)})$.  The cycle $B_{\mathrm{out}}^{(T_c)}$ is
+a subgraph of $O^{(T_p)}$ that bounds a face of $O^{(T_p)}$ 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 $O^{(p)}$.
+exactly a bounded face of $O^{(T_p)}$.

-\emph{Children description.}  The bounded faces of $O^{(p)}$ are
+\emph{Children description.}  The bounded faces of $O^{(T_p)}$ 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-
@@ -926,12 +952,12 @@ decreases depth, terminating at $T_0$.  No cycles can form
 \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
+inner outerplanar graph $O^{(T_p)}$ 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 $O^{(p)}$
+$G$).  This happens, for instance, when $O^{(T_p)}$ has chords or
+cut-vertices that subdivide its inner region, or when $O^{(T_p)}$
 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
+By contrast, if $O^{(T_p)}$ 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}
@@ -1127,8 +1153,8 @@ proper coloring problem on $G$'s bounded faces factors through:
      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 $O^{(p)}$).
+      cycle $B_{\mathrm{out}}^{(T_c)}$ shared between a child and the
+      face of its parent's $O^{(T_p)}$).
 \end{itemize}
 This is the structural setup underlying the chain-pigeonhole
 program for tire treads.