66 lines
6.7 KiB
TeX
66 lines
6.7 KiB
TeX
\relax
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\citation{bauerfeld-nested-tire-duals}
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\citation{birkhoff-reducibility}
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\citation{birkhoff-lewis-chromatic}
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\citation{tutte-chromatic-sums-1973}
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\citation{tutte-algebraic-colorings}
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\citation{tutte-four-colour-conjecture}
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\citation{dvorak-lidicky-cones}
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\citation{heesch-untersuchungen}
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\citation{robertson-sanders-seymour-thomas}
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\citation{robertson-sanders-seymour-thomas}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent }
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\newlabel{def:dual}{{1.3}{2}}
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\newlabel{def:dual-depth}{{1.4}{2}}
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\newlabel{def:dual-component}{{1.5}{2}}
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\newlabel{def:tire-graph}{{1.6}{2}}
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\@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.}}{3}{}\protected@file@percent }
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\newlabel{fig:dual-depth}{{1}{3}}
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\@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 }
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\newlabel{fig:tire-example}{{2}{4}}
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\newlabel{def:medial-tire-graph}{{1.7}{4}}
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\newlabel{rem:tire-counts}{{1.8}{4}}
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\newlabel{prop:no-level-d-pinch}{{1.9}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The medial tire graph for the tire in Figure\nonbreakingspace 2\hbox {}. The chord of $O$ is drawn faintly and omitted before taking the medial graph; medial edges between consecutive outer-boundary edges or consecutive inner-boundary edges are also omitted. Each medial vertex is placed at the midpoint of its corresponding retained tire edge.}}{5}{}\protected@file@percent }
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\newlabel{fig:medial-tire-example}{{3}{5}}
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\citation{bauerfeld-depth}
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\newlabel{lem:tire-component}{{1.10}{6}}
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\citation{bauerfeld-depth}
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\newlabel{thm:tread-partition}{{1.11}{7}}
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\newlabel{rem:tire-component-degenerate}{{1.12}{8}}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.13}{8}}
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\newlabel{thm:inner-dual-outerplanar}{{1.14}{8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\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.}}{9}{}\protected@file@percent }
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\newlabel{fig:inner-dual-disk-case}{{4}{9}}
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\citation{bauerfeld-nested-tire-duals}
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\citation{bauerfeld-nested-tire-duals}
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\newlabel{rem:hamilton-cycle-spoke-only}{{1.15}{10}}
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\newlabel{rem:bridge-case-theta}{{1.16}{10}}
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\newlabel{thm:tread-tree}{{1.17}{10}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\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.}}{11}{}\protected@file@percent }
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\newlabel{fig:inner-dual-annulus-case}{{5}{11}}
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\newlabel{rem:tree-multiple-children}{{1.18}{12}}
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\newlabel{thm:tire-tree-decomposition}{{1.19}{12}}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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\bibcite{birkhoff-reducibility}{4}
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\bibcite{birkhoff-lewis-chromatic}{5}
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\bibcite{tutte-four-colour-conjecture}{6}
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\newlabel{rem:tree-coloring-factorisation}{{1.20}{14}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
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\bibcite{tutte-algebraic-colorings}{7}
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\bibcite{tutte-chromatic-sums-1973}{8}
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\bibcite{heesch-untersuchungen}{9}
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\bibcite{robertson-sanders-seymour-thomas}{10}
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\bibcite{dvorak-lidicky-cones}{11}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{14.69437pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\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 $(a)$.}}{15}{}\protected@file@percent }
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\newlabel{fig:tire-tree-decomposition}{{6}{15}}
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\gdef \@abspage@last{15}
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