\relax 
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Anatomy of a tire annular face connector $T'_{f'}$ in the spoke-only case. \textbf  {Left:} the core $T'_{\mathrm  {ann}} = C_n + M$ (here $\theta (1,3,3)$), the object the menagerie \S  6 formula counts. \textbf  {Right:} $T'_{f'}$ adds one pendant spoke edge for each non-chord boundary vertex; the spoke vertex is a dual vertex of a face of $G$ \emph  {outside} the tire annulus. In a nested-tire setting, that outside face belongs to the next tire over.\relax }}{1}{}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:fiber-setup}{{1}{1}}
\@writefile{toc}{\contentsline {paragraph}{What the fiber count really is.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Special case: $E_S = \emptyset $.}{2}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three spoke configurations $\sigma $ on $C_4$ with four pendant spokes, with cycle vertices labelled $v_0, v_1, v_2, v_3$ and spoke vertices $u_0, u_1, u_2, u_3$. \textbf  {Left:} the constant configuration $\sigma = (1,1,1,1)$ (all spokes red) is realisable with fiber count $N(\sigma ) = 2$ --- the cycle alternates between the two remaining colors in two ways. \textbf  {Middle:} the alternating configuration $\sigma = (1,2,1,2)$ is \emph  {infeasible} ($N(\sigma ) = 0$); the constraints at $v_0$ and $v_1$ force $c(e_0) = 3$, but the constraint at $v_2$ then demands $c(e_1) \in \{2, 3\} \setminus \{c(e_0)\}$ contradicting the $v_1$ constraint. \textbf  {Right:} the non-uniform configuration $\sigma = (1,1,2,2)$ has fiber count $N(\sigma ) = 1$ --- a single cycle coloring extends it.\relax }}{3}{}\protected@file@percent }
\newlabel{fig:fiber-c4}{{2}{3}}
\@writefile{toc}{\contentsline {paragraph}{Total count.}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Distribution over $\sigma $.}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Where this leads.}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Two adjacent tires $T^{\mathrm  {outer}}, T^{\mathrm  {inner}}$ in a common maximal planar $G$, sharing the cycle $\gamma $. The cycle vertex $v \in V(\gamma )$ is a dual vertex for an annular face in \emph  {both} tires. The $G'$-edges at $v$ split into: two cycle edges inside $T'^{\mathrm  {outer}}_{\mathrm  {ann}}$, two cycle edges inside $T'^{\mathrm  {inner}}_{\mathrm  {ann}}$, and (in the spoke-only case) \emph  {one shared spoke edge} $vu$ pointing across the shared boundary. That single spoke edge is simultaneously a spoke of $T'^{\mathrm  {outer}}_{f'}$ and a spoke of $T'^{\mathrm  {inner}}_{f'}$, so any global edge coloring of $G'$ assigns it a single color from both sides at once.\relax }}{5}{}\protected@file@percent }
\newlabel{fig:nesting-compatibility}{{3}{5}}
\gdef \@abspage@last{5}
