coloring_nested_tire_graphs: prove Lemma 1.7 using Lemma 2.6 of plane_depth_sequencing
- Adds a proof sketch to Lemma 1.7 (tire-component lemma). The
outerplanarity step cites Lemma 2.6 of `bauerfeld-pds` (the
Plane Depth Sequencing manuscript), which proves that for any
level source the subgraph induced on a single depth level is
outerplanar. The proof notes that the cited result, originally
stated for an outer-cycle source, extends verbatim to an arbitrary
level source by treating S as the depth-0 set.
- The remainder of the proof: layer containment forces V_{C'} ⊆
L_d ∪ L_{d+1}; a face-by-face boundary analysis shows ∂R_{C'} is
monochromatic in level; connectivity of C' rules out higher genus;
the resulting one or two closed boundary walks give the tire
graph's two boundary parts (with the degenerate case at the BFS
endpoints).
- Adds a thebibliography block with one entry for the cited paper.
The paper grows from 3 to 4 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -5,13 +5,16 @@
|
||||
\newlabel{fig:dual-depth}{{1}{2}}
|
||||
\newlabel{def:tire-graph}{{1.5}{2}}
|
||||
\newlabel{rem:tire-counts}{{1.6}{2}}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{0pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\citation{bauerfeld-pds}
|
||||
\@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{lem:tire-component}{{1.7}{3}}
|
||||
\newlabel{rem:tire-component-degenerate}{{1.8}{3}}
|
||||
\gdef \@abspage@last{3}
|
||||
\bibcite{bauerfeld-pds}{1}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\newlabel{rem:tire-component-degenerate}{{1.8}{4}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{4}
|
||||
|
||||
@@ -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 15:00
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:17
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -210,34 +210,35 @@ Package pdftex.def Info: fig_tire_example.png used on input line 154.
|
||||
|
||||
LaTeX Warning: `h' float specifier changed to `ht'.
|
||||
|
||||
[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] (./paper.aux) )
|
||||
[2 <./fig_dual_depth.png>] [3 <./fig_tire_example.png>] [4] (./paper.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
3005 strings out of 478268
|
||||
41970 string characters out of 5846347
|
||||
338147 words of memory out of 5000000
|
||||
21052 multiletter control sequences out of 15000+600000
|
||||
3006 strings out of 478268
|
||||
41985 string characters out of 5846347
|
||||
339156 words of memory out of 5000000
|
||||
21053 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,7n,76p,625b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/lo
|
||||
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/loc
|
||||
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/loc
|
||||
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/loca
|
||||
l/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/t
|
||||
exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/tex
|
||||
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texli
|
||||
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/
|
||||
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/202
|
||||
2/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022
|
||||
/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/t
|
||||
exmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/tex
|
||||
mf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texm
|
||||
f-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb>
|
||||
Output written on paper.pdf (3 pages, 449333 bytes).
|
||||
69i,8n,76p,625b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</us
|
||||
r/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></usr/
|
||||
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/l
|
||||
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/loc
|
||||
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local
|
||||
/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/t
|
||||
exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texl
|
||||
ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive
|
||||
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/
|
||||
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/20
|
||||
22/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022
|
||||
/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/
|
||||
texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/te
|
||||
xmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (4 pages, 464467 bytes).
|
||||
PDF statistics:
|
||||
89 PDF objects out of 1000 (max. 8388607)
|
||||
51 compressed objects within 1 object stream
|
||||
97 PDF objects out of 1000 (max. 8388607)
|
||||
56 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
11 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -194,6 +194,47 @@ $C$ inside its closed boundary region are exactly the faces of $G$ in
|
||||
$F_{C'}$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}[Proof sketch]
|
||||
By Lemma~2.6 of \cite{bauerfeld-pds} (whose argument, given for an
|
||||
outer-cycle source, extends verbatim to an arbitrary level source $S$
|
||||
by treating $S$ as the depth-$0$ set), the subgraph $G[L_{d'}]$ is
|
||||
outerplanar for each $d' \geq 0$. Since subgraphs of outerplanar
|
||||
graphs are outerplanar, both $G[V_{C'} \cap L_d]$ and
|
||||
$G[V_{C'} \cap L_{d+1}]$ are outerplanar.
|
||||
|
||||
Layer containment (a consequence of the bounded-step property of BFS
|
||||
on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$
|
||||
has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap
|
||||
L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least
|
||||
one vertex in $L_d$.
|
||||
|
||||
It remains to identify the boundary of $R_{C'} := \bigcup_{f \in
|
||||
F_{C'}} f \subseteq |\Pi_G|$. Each edge on the topological boundary
|
||||
$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a
|
||||
face $f' \notin F_{C'}$. Such an $f'$ has dual depth in $\{d-1, d+1\}$:
|
||||
if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a
|
||||
distinct component of $G'_d$, which contradicts the connectivity of
|
||||
$C'$ together with the fact that adjacent depth-$d$ dual vertices belong
|
||||
to the same component. A short case analysis on the level of the third
|
||||
vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have
|
||||
both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both
|
||||
endpoints in $L_{d+1}$. Each connected component of $\partial R_{C'}$
|
||||
is therefore monochromatic in level.
|
||||
|
||||
Since $C'$ is connected and $R_{C'}$ is a connected planar 2-complex
|
||||
of triangles glued along edges, $\partial R_{C'}$ consists of either
|
||||
one closed walk (when $R_{C'}$ is a topological disk) or two closed
|
||||
walks (when $R_{C'}$ is a topological annulus). These walks are
|
||||
simple cycles in $G$ on the respective level sets (with one cycle
|
||||
possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the
|
||||
endpoints of the BFS, giving the degenerate-boundary case of
|
||||
Definition~\ref{def:tire-graph}). Together with the outerplanarity of
|
||||
$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above,
|
||||
these cycles serve as the two boundary parts of $C$, in either order,
|
||||
and the depth-$d$ triangles in $F_{C'}$ tile the closed region between
|
||||
them.
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:tire-component-degenerate}
|
||||
Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
|
||||
@@ -206,4 +247,13 @@ the orientation of the inherited embedding (equivalently, on which side
|
||||
of $C$ contains the rest of $\Pi_G$).
|
||||
\end{remark}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{bauerfeld-pds}
|
||||
E.~Bauerfeld,
|
||||
\emph{Plane Depth Sequencing},
|
||||
manuscript (math-research repository), 2026.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user