coloring_nested_tire_graphs: define inner and outer spokes (Def 1.17)
Adds a new definition partitioning V(T'_{f'}) \ V(f') by geometric
location relative to the face f':
V_out(T'_{f'}) := { v in V(T'_{f'}) \ V(f') : v lies outside the
closure of f' }
= "outer spokes"
V_in(T'_{f'}) := { v in V(T'_{f'}) \ V(f') : v lies inside the
open region f' }
= "inner spokes"
These are well-defined because the boundary walk of f' is V(f') by
definition, so no element of V(T'_{f'}) \ V(f') sits on ∂f'.
In the spoke-only setting (T'_ann = C_{n+m}), the inner spokes of
the inner face are the O-side non-annular dual vertices and the
outer spokes are the source-side non-annular dual vertices (and
vice-versa for the outer face).
Paper stays at 10 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -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}
|
||||
|
||||
@@ -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.
|
||||
|
||||
<notes/fig_facial_dual_choices.png, id=53, 857.75456pt x 341.92744pt>
|
||||
[9]
|
||||
<notes/fig_facial_dual_choices.png, id=56, 857.75456pt x 341.92744pt>
|
||||
File: notes/fig_facial_dual_choices.png Graphic file (type png)
|
||||
<use notes/fig_facial_dual_choices.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
|
||||
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
|
||||
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
|
||||
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb>
|
||||
@@ -266,7 +264,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive
|
||||
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>
|
||||
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
|
||||
|
||||
Binary file not shown.
@@ -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
|
||||
|
||||
Reference in New Issue
Block a user