didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

coloring_nested_tire_graphs: simplify Lemma 1.7 by requiring S on the outer face

Browse Source 
Pins Π_G at the start to be an embedding placing S on the outer face;
such an embedding exists for any single-vertex source.  This collapses
the two-embedding split in the previous proof (one for Lemma 2.6,
another for the topological analysis of R_{C'}) into a single
embedding throughout, and removes the "in either order" ambiguity for
B_out and B_in:

- B_out = G[V_{C'} ∩ L_d]: the boundary of R_{C'} closer to S.
- B_in  = G[V_{C'} ∩ L_{d+1}]: the boundary farther from S.

The outerplanarity step now cites Lemma 2.6 of [bauerfeld-pds]
directly (no embedding switch).  The "tire structure" step pins the
orientation by S's position on the outer face.

Remark 1.9 (degenerate cases) updated: the orientation ambiguity is
gone, so we state the d=0 case has degenerate B_out and the d=D_max
case has degenerate B_in.

(R1) and (R2) remain — they are graph-theoretic and unaffected by
embedding choice (for 3-connected planar graphs the embedding is
essentially unique by Whitney's theorem, so changing the outer face
cannot untangle pinches or merge multi-hole topology).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 15:36:26 -04:00
parent 688b944f95
commit 1ac80aa5cf
3 changed files with 48 additions and 49 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_graphs/paper.log
+14 -13
View File
@@ -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 15:31 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:36
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -215,7 +215,7 @@ LaTeX Warning: `h' float specifier changed to `ht'.
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
3007 strings out of 478268 3007 strings out of 478268
42001 string characters out of 5846347 42001 string characters out of 5846347
344166 words of memory out of 5000000 345166 words of memory out of 5000000
21054 multiletter control sequences out of 15000+600000 21054 multiletter control sequences out of 15000+600000
475666 words of font info for 53 fonts, out of 8000000 for 9000 475666 words of font info for 53 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191 1302 hyphenation exceptions out of 8191
@@ -227,18 +227,19 @@ usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></u
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr
/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/l /local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/l
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/loc ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/ al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/
texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/tex texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/tex
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texl live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texliv
ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texliv e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive
e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/ /2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2 022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/202
022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/202 2/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022
2/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb> /texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/t
Output written on paper.pdf (5 pages, 477312 bytes). exmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (5 pages, 486337 bytes).
PDF statistics: PDF statistics:
100 PDF objects out of 1000 (max. 8388607) 105 PDF objects out of 1000 (max. 8388607)
58 compressed objects within 1 object stream 61 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000) 0 named destinations out of 1000 (max. 500000)
11 words of extra memory for PDF output out of 10000 (max. 10000000) 11 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/coloring_nested_tire_graphs/paper.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_graphs/paper.tex
+34 -36
View File
@@ -174,8 +174,10 @@ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
\begin{lemma}[Tire-component lemma] \begin{lemma}[Tire-component lemma]
\label{lem:tire-component} \label{lem:tire-component}
Let $G$ be a maximal planar graph with fixed embedding $\Pi_G$ and let Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level
$S \subseteq V(G)$ be a level source. For $d \geq 0$, let source. Fix a plane embedding $\Pi_G$ of $G$ in which $S$ lies on the
outer face (such an embedding exists for any planar graph and any
single-vertex source). For $d \geq 0$, let
\[ \[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr] G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\] \]
@@ -195,21 +197,20 @@ Assume:
\item[\emph{(R2)}] $R_{C'}$ has at most two boundary components. \item[\emph{(R2)}] $R_{C'}$ has at most two boundary components.
\end{itemize} \end{itemize}
Then $C$, with the inherited embedding, is a tire graph in the sense of Then $C$, with the inherited embedding, is a tire graph in the sense of
Definition~\ref{def:tire-graph}: its two boundary parts Definition~\ref{def:tire-graph}. Its outer boundary
$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ are the level-$d$ subgraph $B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
$G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$; its inner boundary
$G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of $B_{\mathrm{in}}$ is the side farther from $S$, namely the
$C$ inside the closed boundary region are exactly the faces of $G$ in level-$(d+1)$ subgraph $G[V_{C'} \cap L_{d+1}]$; and the triangular
$F_{C'}$. faces of $C$ inside the closed boundary region are exactly the faces of
$G$ in $F_{C'}$.
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
\emph{Outerplanarity of the two level parts.} Since $S$ is a single \emph{Outerplanarity of the two level parts.} By construction $S$
vertex, choose a plane embedding of $G$ with $S$ on the outer face. lies on the outer face of $\Pi_G$, so Lemma~2.6 of \cite{bauerfeld-pds}
By Lemma~2.6 of \cite{bauerfeld-pds} applied with this embedding and applies directly with $(G, \Pi_G, S)$, giving that $G[L_{d'}]$ is
source $S$, $G[L_{d'}]$ is outerplanar for each $d' \geq 0$; outerplanar for each $d' \geq 0$. Subgraphs of outerplanar graphs are
outerplanarity is a graph property, so the conclusion is independent
of the embedding choice. Subgraphs of outerplanar graphs are
outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
both outerplanar. both outerplanar.
@@ -265,21 +266,18 @@ the number of boundary components. Hypothesis (R2) gives $n \leq 2$,
so $R_{C'}$ is either a closed disk ($n = 1$) or a closed annulus so $R_{C'}$ is either a closed disk ($n = 1$) or a closed annulus
($n = 2$). ($n = 2$).
\emph{Tire structure.} In the annulus case ($n = 2$), the two \emph{Tire structure.} Because $S$ lies on the outer face of $\Pi_G$,
boundary cycles are simple cycles on $L_d$ and $L_{d+1}$ respectively the level-$d$ vertices are closer to $S$ in $\Pi_G$ than the
(by the previous two paragraphs). These are the cycles bounding the level-$(d+1)$ vertices; in either the annulus or disk case the
two outerplanar subgraphs $G[V_{C'} \cap L_d]$ and boundary cycle on the $L_d$ side is the boundary of $R_{C'}$ facing
$G[V_{C'} \cap L_{d+1}]$ in $\Pi_G$, and they meet the $S$ (the ``outer'' boundary), and the $L_{d+1}$ side is the boundary
tire-graph definition with $B_{\mathrm{out}} \in \{$ those cycles $\}$ facing the interior (the ``inner'' boundary). This identifies
in either order. In the disk case ($n = 1$), the unique boundary $B_{\mathrm{out}} = G[V_{C'} \cap L_d]$ and $B_{\mathrm{in}} =
cycle lies on one of the two levels, and the other level set of G[V_{C'} \cap L_{d+1}]$. In the disk case ($n = 1$) one of the two
$V_{C'}$ is either empty or a single interior vertex of the disk level sets is a single vertex (the BFS endpoint at $d = 0$ with
(the BFS endpoint). When it is a single vertex this is the $S = \{v_0\}$, or symmetrically at $d = D_{\max}$ where the inner
degenerate-boundary case of Definition~\ref{def:tire-graph}; the side collapses to a deepest vertex), giving the degenerate-boundary
remaining case ($V_{C'} \cap L_{d+1} = \emptyset$, which arises at case of Definition~\ref{def:tire-graph}.
$d = D_{\max}$) is excluded by interpreting one boundary part as a
degenerate single vertex on $L_{d+1}$ (taken empty by convention,
which we omit here).
The triangular faces inside the closed boundary region of $C$ are by The triangular faces inside the closed boundary region of $C$ are by
construction the depth-$d$ faces in $F_{C'}$, and the edges of $C$ are construction the depth-$d$ faces in $F_{C'}$, and the edges of $C$ are
@@ -291,13 +289,13 @@ $V_{C'} \cap L_{d+1}$ that bound a face of $F_{C'}$.
\begin{remark} \begin{remark}
\label{rem:tire-component-degenerate} \label{rem:tire-component-degenerate}
Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
degenerate. For instance, at $d = 0$ with single-vertex source degenerate. At $d = 0$ with single-vertex source $S = \{v_0\}$ the
$S = \{v_0\}$ the unique component of $G'_0$ has unique component of $G'_0$ has $V_{C'} \cap L_0 = \{v_0\}$ as the
$V_{C'} \cap L_0 = \{v_0\}$ --- the degenerate boundary --- and degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the
$V_{C'} \cap L_1$ a cycle (the link of $v_0$ in $G$). Which of the two link of $v_0$ in $G$) as the inner boundary. Symmetrically, at
parts is $B_{\mathrm{out}}$ and which is $B_{\mathrm{in}}$ depends on $d = D_{\max}$, $V_{C'} \cap L_{D_{\max}+1} = \emptyset$ degenerates
the orientation of the inherited embedding (equivalently, on which side to a single deepest vertex serving as the \emph{inner} boundary, with
of $C$ contains the rest of $\Pi_G$). the level-$D_{\max}$ cycle as the outer boundary.
\end{remark} \end{remark}
\begin{remark} \begin{remark}