diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index b9a54c6..27d2a3b 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -20,17 +20,18 @@ \newlabel{rem:tire-component-degenerate}{{1.11}{8}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} \newlabel{prop:edge-vertex-bijection}{{1.13}{8}} +\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} +\newlabel{def:tire-annular-subgraph}{{1.15}{9}} +\newlabel{def:tire-annular-face-connector}{{1.16}{9}} +\newlabel{def:spokes}{{1.17}{9}} +\newlabel{rem:facial-dual-spoke-only}{{1.18}{9}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} -\newlabel{def:tire-annular-subgraph}{{1.15}{9}} -\newlabel{def:tire-annular-face-connector}{{1.16}{9}} -\newlabel{rem:facial-dual-spoke-only}{{1.17}{9}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } \newlabel{fig:facial-dual-choices}{{5}{10}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } \gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index baa2443..2fd8af9 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/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) 25 MAY 2026 23:06 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 23:14 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -228,28 +228,26 @@ LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 575. - +[9] + File: notes/fig_facial_dual_choices.png Graphic file (type png) Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -628. +654. (pdftex.def) Requested size: 360.0pt x 143.50418pt. - -LaTeX Warning: `h' float specifier changed to `ht'. - -[9] [10 <./notes/fig_facial_dual_choices.png>] (./paper.aux) + [10 <./notes/fig_facial_dual_choices.png>] (./paper.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: - 3041 strings out of 478268 - 43057 string characters out of 5846347 - 344281 words of memory out of 5000000 - 21084 multiletter control sequences out of 15000+600000 + 3042 strings out of 478268 + 43069 string characters out of 5846347 + 344292 words of memory out of 5000000 + 21085 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,8n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s + 69i,14n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -266,7 +264,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (10 pages, 813184 bytes). +Output written on paper.pdf (10 pages, 814471 bytes). PDF statistics: 128 PDF objects out of 1000 (max. 8388607) 73 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 9aa98c8..4278bea 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 3259206..543b7cf 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -604,6 +604,32 @@ i.e.\ the subgraph of $G'$ on the closed $G'$-neighborhood of $V(f')$ together with every $G'$-edge incident to $V(f')$. \end{definition} +\begin{definition}[Inner and outer spokes] +\label{def:spokes} +With $T'_{f'}$ as in +Definition~\ref{def:tire-annular-face-connector}, regard $f'$ as an +open region of $|\Pi_G|$ and write $\overline{f'}$ for its closure. +The vertices of $V(T'_{f'}) \setminus V(f')$ lie in $|\Pi_G| \setminus +\overline{f'}$ or in $f'$ (never on $\partial f'$, since the boundary +walk of $f'$ is by definition the set $V(f')$). Partition +\[ + V(T'_{f'}) \setminus V(f') \;=\; + V_{\mathrm{out}}(T'_{f'}) \,\sqcup\, V_{\mathrm{in}}(T'_{f'}) +\] +where +\begin{align*} + V_{\mathrm{out}}(T'_{f'}) &:= \{\, v \in V(T'_{f'}) \setminus V(f') + \;:\; v \notin \overline{f'} \,\}, \\ + V_{\mathrm{in}}(T'_{f'}) &:= \{\, v \in V(T'_{f'}) \setminus V(f') + \;:\; v \in f' \,\}. +\end{align*} +The elements of $V_{\mathrm{out}}(T'_{f'})$ are the \emph{outer +spokes} of $T'_{f'}$ (vertices not in $V(f')$ that lie outside the +region bounded by $f'$); the elements of $V_{\mathrm{in}}(T'_{f'})$ +are the \emph{inner spokes} of $T'_{f'}$ (vertices not in $V(f')$ +that lie inside the region bounded by $f'$). +\end{definition} + \begin{remark} \label{rem:facial-dual-spoke-only} In the spoke-only setting of