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

papers: move tire-component lemma + tread partition theorem to foundational paper

Browse Source 
Moved from coloring_nested_tire_dual_graphs/ TO coloring_nested_tire_graphs/:
  - Proposition (Source-side simple-cycle property) → now 1.7
  - Lemma (Tire-component lemma) → now 1.8
  - Theorem (Tire treads partition the bounded faces) → now 1.9
  - Remark (boundaries-may-be-degenerate) → now 1.10
  - Remark (no extra hypotheses needed) → now 1.11

These are foundational structural results about tire-graph
decompositions induced by a level source, not specifically about
the partial tire dual D(T) or coloring.  Belongs in the
foundational paper.

Updates:
  - Internal \cite[Definition~1.5]{bauerfeld-nested-tires} inside
    the moved blocks → local \ref{def:tire-graph}.
  - Foundational paper abstract rewritten to highlight the
    tire-component lemma and tread partition as the main results.
  - Dual paper abstract trimmed: no longer claims the tire-component
    lemma as its own contribution.
  - Dual paper intro citation list adds bullets for the moved
    lemma (\cite[Lemma~1.8]) and theorem (\cite[Theorem~1.9]).
  - No external references to the moved items inside the dual paper.

Page counts:
  - Foundational: 3 → 7 pages.
  - Dual: 9 → 7 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 01:13:25 -04:00
parent 401dc86853
commit 9185d626af
8 changed files with 425 additions and 406 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_dual_graphs/paper.aux
+22 -30
View File
@@ -6,38 +6,30 @@
\citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\newlabel{def:partial-tire-dual}{{1.1}{1}} \citation{bauerfeld-nested-tires}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } \newlabel{def:partial-tire-dual}{{1.1}{2}}
\newlabel{fig:partial-tire-dual-example}{{1}{2}}
\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} \newlabel{prop:partial-tire-dual-structure}{{1.2}{2}}
\citation{bauerfeld-nested-tires} \newlabel{prop:edge-vertex-bijection}{{1.3}{2}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{3}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{3}{}\protected@file@percent }
\newlabel{fig:partial-tire-dual-bridge}{{2}{3}} \newlabel{fig:partial-tire-dual-example}{{1}{3}}
\newlabel{prop:no-level-d-pinch}{{1.3}{3}} \newlabel{rem:edge-vertex-corollary}{{1.4}{3}}
\citation{bauerfeld-nested-tires}
\newlabel{lem:tire-component}{{1.4}{4}}
\citation{bauerfeld-depth}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-depth}
\citation{bauerfeld-nested-tires}
\newlabel{thm:tread-partition}{{1.5}{6}}
\newlabel{rem:tire-component-degenerate}{{1.6}{6}}
\citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires}
\newlabel{rem:tire-no-extra-hypotheses}{{1.7}{7}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{4}{}\protected@file@percent }
\newlabel{prop:edge-vertex-bijection}{{1.8}{7}} \newlabel{fig:partial-tire-dual-bridge}{{2}{4}}
\newlabel{rem:edge-vertex-corollary}{{1.9}{7}} \newlabel{def:tire-annular-subgraph}{{1.5}{4}}
\newlabel{def:tire-annular-subgraph}{{1.10}{7}} \newlabel{def:tire-annular-face-connector}{{1.6}{4}}
\newlabel{def:tire-annular-face-connector}{{1.11}{8}} \newlabel{def:spokes}{{1.7}{5}}
\newlabel{def:spokes}{{1.12}{8}} \newlabel{rem:facial-dual-spoke-only}{{1.8}{5}}
\newlabel{rem:facial-dual-spoke-only}{{1.13}{8}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{5}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{8}{}\protected@file@percent } \newlabel{sec:latin-conjecture}{{2}{5}}
\newlabel{sec:latin-conjecture}{{2}{8}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\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 $u_4, u_5$).}}{6}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\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 $u_4, u_5$).}}{9}{}\protected@file@percent } \newlabel{fig:facial-dual-choices}{{3}{6}}
\newlabel{fig:facial-dual-choices}{{3}{9}} \newlabel{conj:latin}{{2.1}{6}}
\newlabel{conj:latin}{{2.1}{9}} \newlabel{conj:chain-latin}{{2.2}{6}}
\newlabel{conj:chain-latin}{{2.2}{9}}
\bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-depth}{1}
\bibcite{bauerfeld-nested-tires}{2} \bibcite{bauerfeld-nested-tires}{2}
\newlabel{tocindent-1}{0pt} \newlabel{tocindent-1}{0pt}
@@ -45,5 +37,5 @@
\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent }
\gdef \@abspage@last{10} \gdef \@abspage@last{7}
papers/coloring_nested_tire_dual_graphs/paper.log
+49 -48
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) 27 MAY 2026 01:09 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:13
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -195,32 +195,34 @@ e
<fig_partial_tire_dual.png, id=22, 657.657pt x 546.54187pt> <fig_partial_tire_dual.png, id=22, 657.657pt x 546.54187pt>
File: fig_partial_tire_dual.png Graphic file (type png) File: fig_partial_tire_dual.png Graphic file (type png)
<use fig_partial_tire_dual.png> <use fig_partial_tire_dual.png>
Package pdftex.def Info: fig_partial_tire_dual.png used on input line 143. Package pdftex.def Info: fig_partial_tire_dual.png used on input line 151.
(pdftex.def) Requested size: 280.79956pt x 233.36552pt. (pdftex.def) Requested size: 280.79956pt x 233.36552pt.
<fig_partial_tire_dual_bridge.png, id=24, 780.96768pt x 522.15076pt>
File: fig_partial_tire_dual_bridge.png Graphic file (type png)
<use fig_partial_tire_dual_bridge.png>
Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 1
58.
(pdftex.def) Requested size: 306.0022pt x 204.59406pt.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
[2 <./fig_partial_tire_dual.png>] [3 <./fig_partial_tire_dual_bridge.png>] <fig_partial_tire_dual_bridge.png, id=24, 780.96768pt x 522.15076pt>
[4] [5] [6] [7] File: fig_partial_tire_dual_bridge.png Graphic file (type png)
<notes/fig_facial_dual_choices.png, id=52, 857.75456pt x 341.92744pt> <use fig_partial_tire_dual_bridge.png>
Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 1
66.
(pdftex.def) Requested size: 306.0022pt x 204.59406pt.
LaTeX Warning: `h' float specifier changed to `ht'.
[2] [3 <./fig_partial_tire_dual.png>] [4 <./fig_partial_tire_dual_bridge.png>]
<notes/fig_facial_dual_choices.png, id=40, 857.75456pt x 341.92744pt>
File: notes/fig_facial_dual_choices.png Graphic file (type png) File: notes/fig_facial_dual_choices.png Graphic file (type png)
<use notes/fig_facial_dual_choices.png> <use notes/fig_facial_dual_choices.png>
Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line
626. 363.
(pdftex.def) Requested size: 360.0pt x 143.50418pt. (pdftex.def) Requested size: 360.0pt x 143.50418pt.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
[8]
Overfull \hbox (68.454pt too wide) detected at line 677 Overfull \hbox (68.454pt too wide) detected at line 414
\OMS/cmsy/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B[]; O\OT1/cmr/m/n/10 ) := \OMS/cmsy/m/n/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 B[]; O\OT1/cmr/m/n/10 ) :=
[] \OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 ( [] \OML/cmm/m/it/10 ^^[ \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (
\OML/cmm/m/it/10 B[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! f\OT1/cmr/m/n/10 1\OML \OML/cmm/m/it/10 B[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! f\OT1/cmr/m/n/10 1\OML
@@ -231,43 +233,42 @@ m/m/it/10 ; \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 3\OMS/cmsy/m/n/
\OML/cmm/m/it/10 O\OT1/cmr/m/n/10 ) []\OML/cmm/m/it/10 : \OML/cmm/m/it/10 O\OT1/cmr/m/n/10 ) []\OML/cmm/m/it/10 :
[] []
[9 <./notes/fig_facial_dual_choices.png>] [10] (./paper.aux) ) [5] [6 <./notes/fig_facial_dual_choices.png>] [7] (./paper.aux) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
3037 strings out of 478268 3029 strings out of 478268
42864 string characters out of 5846347 42706 string characters out of 5846347
346314 words of memory out of 5000000 346258 words of memory out of 5000000
21080 multiletter control sequences out of 15000+600000 21073 multiletter control sequences out of 15000+600000
476532 words of font info for 56 fonts, out of 8000000 for 9000 475834 words of font info for 54 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191 1302 hyphenation exceptions out of 8191
69i,14n,76p,1079b,300s stack positions out of 10000i,1000n,20000p,200000b,200000s 69i,14n,76p,1079b,288s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texli </usr/local/te
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texliv xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/tex
e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texliv live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/tex
e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texl
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/ ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texli
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/20 ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive
22/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022 /2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2
/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/t 022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/202
exmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/tex 2/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/
mf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/te
-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf-di xmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf
st/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/ -dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-di
fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fon st/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/
ts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/ fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fon
type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/t ts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/font
ype1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ s/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/
e1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1 type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/ty
/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p pe1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type
ublic/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub 1/public/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/
lic/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
ic/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public ublic/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ lic/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
amsfonts/symbols/msam10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub ic/amsfonts/symbols/msam10.pfb>
lic/amsfonts/symbols/msbm10.pfb> Output written on paper.pdf (7 pages, 583446 bytes).
Output written on paper.pdf (10 pages, 614303 bytes).
PDF statistics: PDF statistics:
159 PDF objects out of 1000 (max. 8388607) 145 PDF objects out of 1000 (max. 8388607)
94 compressed objects within 1 object stream 85 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000) 0 named destinations out of 1000 (max. 500000)
16 words of extra memory for PDF output out of 10000 (max. 10000000) 16 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/coloring_nested_tire_dual_graphs/paper.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_dual_graphs/paper.tex
+20 -283
View File
@@ -46,17 +46,17 @@
\begin{abstract} \begin{abstract}
This is a follow-up to \cite{bauerfeld-nested-tires}, which This is a follow-up to \cite{bauerfeld-nested-tires}, which
establishes the basic vocabulary of tire graphs $T$ and dual establishes the basic vocabulary of tire graphs $T$ and dual
depth. Building on those definitions, we define the depth, along with the tire-component lemma and the tire-tread
\emph{partial tire dual} $D(T)$ and analyse its structure in the partition theorem. Building on those structural results, we
spoke-only case (a corona graph $C_{n+m} \circ K_1$), prove the define the \emph{partial tire dual} $D(T)$ and analyse its
tire-component lemma structure in the spoke-only case (a corona graph $C_{n+m} \circ
exhibiting every BFS-level component as a tire graph, give an K_1$), give an edge-vertex coloring bijection that reduces
edge-vertex coloring bijection that reduces counting proper counting proper $3$-edge-colorings of $D(T)$ to counting proper
$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings $3$-vertex-colorings of a cycle, and develop the
of a cycle, and develop the tire-annular-subgraph, face-connector, tire-annular-subgraph, face-connector, and inner/outer-spoke
and inner/outer-spoke structures in $G'$. A concluding section structures in $G'$. A concluding section records a
records a Latin-substructure conjecture for chain-pigeonhole Latin-substructure conjecture for chain-pigeonhole compatibility
compatibility of adjacent tires. of adjacent tires.
\end{abstract} \end{abstract}
\maketitle \maketitle
@@ -87,7 +87,15 @@ particular we use, without restating, the notions of:
with outer/inner boundaries and annular edges with outer/inner boundaries and annular edges
(\cite[Definition~1.5]{bauerfeld-nested-tires}); (\cite[Definition~1.5]{bauerfeld-nested-tires});
\item face/edge counts \item face/edge counts
(\cite[Remark~1.6]{bauerfeld-nested-tires}). (\cite[Remark~1.6]{bauerfeld-nested-tires});
\item the \emph{tire-component lemma}
(\cite[Lemma~1.8]{bauerfeld-nested-tires}), which exhibits
each connected component of $G'_d$ as a tire graph
whose tire tread is the union of its depth-$d$ faces;
\item the \emph{tire-tread partition theorem}
(\cite[Theorem~1.9]{bauerfeld-nested-tires}), which shows
the tire treads from a level source partition the bounded
faces of $G$.
\end{itemize} \end{itemize}
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
@@ -209,277 +217,6 @@ cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n +
and there are also $n + m$ leaves attached one-per-cycle-vertex. and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof} \end{proof}
\begin{proposition}[Source-side simple-cycle property]
\label{prop:no-level-d-pinch}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let
$C'$ be a connected component of $G'_d$ such that $v$ is incident to
some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident
to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$.
Equivalently: for any such component, the source-side boundary of
$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$).
\end{proposition}
\begin{proof}
Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$
lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices
in $G$ differ in level by at most $1$, so a face at $v$ has depth
exactly $d$ iff both other vertices have level $\geq d$, and depth
$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between
the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of
$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick
$p$ in one such gap and $q$ in another.
The BFS ball $G[L_{<d}]$ is connected, so there exists a simple path
$P$ in $G[L_{<d}]$ from $p$ to $q$. Define the closed walk
\[
W \;:=\; v \to p \to P \to q \to v.
\]
Every vertex of $P$ lies in $L_{<d}$, while $\ell(v) = d$, so $v$ is
distinct from every vertex of $P$; $P$ is simple, so its internal
vertices are distinct; and $p \neq q$ since they lie in different gaps.
Hence $W$ is a simple cycle in $G$.
By the Jordan curve theorem, the planar embedding of $W$ divides $\Pi_G$
into two regions. In $v$'s rotation, the edges $v-p$ and $v-q$ lie at
two specific positions, and they split the rotation into two arcs;
each arc lies in one of the two regions determined by $W$. By choice
of $p, q$, the two arcs of depth-$d$ faces at $v$ in $F_{C'}$ lie in
different regions of $W$ (i.e., one arc on each side).
Since $C'$ is connected in $G'$ and contains depth-$d$ faces in both
arcs, there is a dual path $f_1, f_2, \ldots, f_k$ in $G'_d$ with
$f_1, f_k \in F_{C'}$ incident to $v$ in different arcs, and with the
intermediate faces $f_2, \ldots, f_{k-1}$ not incident to $v$ (a
shortest such dual path). Consecutive faces $f_i, f_{i+1}$ share an
edge $e_i$ of $G$; for $i \geq 2$, both endpoints of $e_i$ lie in
$L_{\geq d}$ (since neither $f_i$ nor $f_{i+1}$ is incident to $v$,
all six vertices of these two triangles lie in $L_{\geq d}$). In
particular, $e_i$ shares no endpoint with $W$ except possibly $v$ ---
and $v$ is excluded from $f_2, \ldots, f_{k-1}$.
A planar edge with neither endpoint on a simple closed planar curve
$W$ has both of its incident faces on the same side of $W$. Applying
this to each $e_i$ ($i \geq 2$) inductively: starting from $f_2$ on
the same side of $W$ as $f_1$ (their shared edge $e_1 = w-w'$ opposite
to $v$ in $f_1$ has $w, w' \in L_{\geq d}$ and hence is not on $W$),
the path $f_2 \to f_3 \to \cdots \to f_{k-1} \to f_k$ stays on one
side of $W$.
But $f_1$ and $f_k$ lie on different sides of $W$ (by construction),
contradicting the conclusion that the entire path lies on one side.
\end{proof}
\begin{lemma}[Tire-component lemma]
\label{lem:tire-component}
Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level
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
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\]
be the inner-dual subgraph on dual vertices of dual depth $d$, and let
$C'$ be a connected component of $G'_d$. Write
$F_{C'} := \{f : d_f \in V(C')\}$,
$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit
its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of
\cite[Definition~1.5]{bauerfeld-nested-tires}. Its outer boundary
$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
boundary closed walk of $O$ in the inherited embedding (a simple cycle
when $O$ is $2$-connected, a non-simple closed walk in general). The
triangular faces of $C$ inside the closed boundary region are exactly
the faces of $G$ in $F_{C'}$.
\end{lemma}
\begin{proof}
\emph{Outerplanarity of the two level parts.} By construction $S$
lies on the outer face of $\Pi_G$, so the outerplanarity lemma of
\cite{bauerfeld-depth} applies directly with $(G, \Pi_G, S)$, giving
that $G[L_{d'}]$ is
outerplanar for each $d' \geq 0$. Subgraphs of outerplanar graphs are
outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
both outerplanar.
\emph{Layer containment.} Each $f \in F_{C'}$ has at least one vertex
at level $d$, and adjacent vertices in $G$ differ in level by at most
$1$; combined with $\delta_G(d_f) = d$, this forces
$V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup
L_{d+1}$, and $C$ has vertex partition
$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
\emph{Boundary edges are monochromatic in level.} Each edge $e$ on
$\partial R_{C'}$ separates a face $f \in F_{C'}$ from a face
$f' \notin F_{C'}$. Because $f$ and $f'$ share the edge $e$, their
dual vertices are adjacent in $G'$; if both had depth $d$ they would
lie in the same component of $G'_d$, contradicting $d_{f} \in C'$ and
$d_{f'} \notin C'$. Hence $\delta_G(d_{f'}) \neq d$; combined with
the bounded-step property of $\delta$ across $G'$-adjacent faces,
$\delta_G(d_{f'}) \in \{d-1, d+1\}$.
\begin{itemize}
\item If $\delta_G(d_{f'}) = d - 1$, the third vertex $w$ of
$f' = \{u, v, w\}$ (where $u, v$ are the endpoints of $e$) has
$\ell(w) = d - 1$. Each of $u, v$ has $\ell \in \{d, d+1\}$
(from $V(f) \subseteq L_d \cup L_{d+1}$) and is adjacent to $w$,
forcing $\ell(u), \ell(v) \in \{d-2, d-1, d\} \cap \{d, d+1\} =
\{d\}$.
\item If $\delta_G(d_{f'}) = d + 1$, then all three vertices of $f'$
lie in $L_{\geq d+1}$, so in particular $\ell(u) = \ell(v) =
d + 1$.
\end{itemize}
Each connected boundary component thus carries a single type at every
edge: any vertex on a boundary component has two boundary edges
incident to it (by R1, see below), both of the same type, so its
level is fixed. Therefore each boundary component of $\partial R_{C'}$
is monochromatic in level.
\emph{Boundary structure.} Each connected component of
$\partial R_{C'}$ traces a closed walk in $G$ that, by the
monochromaticity above, lies entirely in $L_d$ or entirely in
$L_{d+1}$. By Proposition~\ref{prop:no-level-d-pinch}, the depth-$d$
faces of $F_{C'}$ at any $v \in L_d \cap V_{C'}$ form a single
contiguous arc in $v$'s rotation, so the source-side boundary walk
visits each $L_d$-vertex of $V_{C'}$ exactly once: it is a simple
cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
may split into multiple arcs of $v$'s rotation; this corresponds
exactly to $v$ being a cut-vertex of $O$, and the inner-side
boundary walk visits $v$ correspondingly many times --- which is
already accommodated by \cite[Definition~1.5]{bauerfeld-nested-tires} (where
$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
necessarily a simple cycle).
\emph{Outer boundary.} Because $S$ lies on the outer face of $\Pi_G$,
the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$
in the embedding. In the inherited embedding of $C$, the unique
unbounded face is the merged region containing the rest of $\Pi_G$
outside $R_{C'}$ on the $S$ side, so its boundary --- a simple cycle
on $L_d$ (or a single vertex when $V_{C'} \cap L_d = \{v_0\}$, the
$d = 0$ case) --- serves as $B_{\mathrm{out}}$. We set
$B_{\mathrm{out}} := G[V_{C'} \cap L_d]$ if this is a cycle, and
the single vertex $\{v_0\}$ in the degenerate case.
\emph{Inner outerplanar graph.} By the outerplanarity lemma of
\cite{bauerfeld-depth}, $G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O :=
G[V_{C'} \cap L_{d+1}]$. The boundary curve(s) of $R_{C'}$ on the
$L_{d+1}$ side are exactly the boundary of $O$'s outer face in the
inherited embedding; this outer-face boundary is a single closed walk
that traces around $O$ from the outside, traversing any bridge edge
twice and visiting cut-vertices multiple times. This walk is the
inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s
internal structure is needed: when $R_{C'}$ has more than two
boundary components in the surface-classification sense (i.e.\
several disjoint simple cycles on $L_{d+1}$), these correspond
precisely to the multiple connected components or bridge crossings of
$O$, and the outer-face boundary closed walk of $O$ captures them
collectively.
\emph{Tire structure.} The triangular faces of $C$ inside the closed
boundary region are by construction the depth-$d$ faces in $F_{C'}$,
and the edges of $C$ are $E(B_{\mathrm{out}}) \cup E(O) \cup
E_{\mathrm{ann}}$ where $E_{\mathrm{ann}}$ are the edges of $G$
between $V_{C'} \cap L_d$ and $V_{C'} \cap L_{d+1}$ that bound a face
of $F_{C'}$.
\end{proof}
\begin{theorem}[Tire treads partition the bounded faces]
\label{thm:tread-partition}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
let $S \subseteq V(G)$ be a level source lying on the outer face.
For each $d \ge 0$ and each connected component $C'$ of $G'_d$, let
$T^{(d, C')}$ denote the tire graph supplied by
Lemma~\ref{lem:tire-component}, with tire tread
$R_{C'} \subseteq |\Pi_G|$. Then the collection of treads
\[
\mathcal{R}(G, S) \;:=\;
\bigl\{\, R_{C'} \,:\, d \ge 0,\;
C' \text{ a connected component of } G'_d \,\bigr\}
\]
partitions the bounded part of $|\Pi_G|$:
\begin{enumerate}
\item[(i)] every bounded face $f$ of $G$ is contained in exactly
one tread $R_{C'} \in \mathcal{R}(G, S)$;
\item[(ii)] distinct treads in $\mathcal{R}(G, S)$ have disjoint
interiors and may share only boundary edges or vertices.
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{Existence and uniqueness.} Each bounded face $f \in F(G)$
has a uniquely-defined dual depth $\delta_G(d_f) \in \mathbb{Z}_{\ge
0}$, so the dual vertex $d_f$ lies in $G'_d$ for $d =
\delta_G(d_f)$ and in no other $G'_{d'}$. Within $G'_d$, the
vertex $d_f$ belongs to exactly one connected component $C'$. By
Lemma~\ref{lem:tire-component}, $F_{C'}$ is precisely the set of
faces $f' \in F(G)$ with $d_{f'} \in V(C')$; in particular $f \in
F_{C'}$, hence $f \subseteq R_{C'}$.
For any other tread $R_{C''} \in \mathcal{R}(G, S)$, the
component $C''$ is either at a different depth $d' \ne d$ (in
which case $F_{C''}$ consists of depth-$d'$ faces and $f \notin
F_{C''}$) or at depth $d$ but a different component $C'' \ne C'$
(in which case the two components are vertex-disjoint in $G'_d$,
so again $f \notin F_{C''}$). In both cases $f \notin R_{C''}$
(more precisely, $f$ is not one of the triangular faces of $G$ in
$F_{C''}$, so $f$'s interior is not contained in $R_{C''}$).
\emph{Disjoint interiors.} Each tread $R_{C'}$ is the union of
its triangular faces $F_{C'} \subseteq F(G)$; distinct treads
correspond to disjoint $F_{C'}$ (by the argument above), and the
interiors of distinct $G$-faces are disjoint. Hence interiors of
distinct treads are disjoint.
\emph{Coverage.} Conversely, every bounded $f \in F(G)$ has $d_f
\in V(G')$ with some dual depth $d$, and thus lies in $R_{C'}$
where $C'$ is its component of $G'_d$. So $\bigcup_{R \in
\mathcal{R}(G, S)} R$ contains every bounded face of $G$.
\end{proof}
\begin{remark}
\label{rem:tire-component-degenerate}
Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
degenerate. At $d = 0$ with single-vertex source $S = \{v_0\}$ the
unique component of $G'_0$ has $V_{C'} \cap L_0 = \{v_0\}$ as the
degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the
link of $v_0$ in $G$) as the inner boundary. Symmetrically, at
$d = D_{\max}$, $V_{C'} \cap L_{D_{\max}+1} = \emptyset$ degenerates
to a single deepest vertex serving as the \emph{inner} boundary, with
the level-$D_{\max}$ cycle as the outer boundary.
\end{remark}
\begin{remark}
\label{rem:tire-no-extra-hypotheses}
Two structural features of $R_{C'}$ that might at first appear to
obstruct the tire-graph conclusion are both already accommodated by
\cite[Definition~1.5]{bauerfeld-nested-tires}:
\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may
have the faces of $F_{C'}$ incident to it split into two or more
arcs in $v$'s rotation in $\Pi_G$, separated by faces of higher
depth. This corresponds exactly to $v$ being a cut-vertex of
$O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk
$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each
arc. No additional hypothesis is needed.
\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses
several disjoint depth-$>d$ sub-regions, the inner outerplanar graph
$O$ captures the multi-hole structure as a disconnected or
non-$2$-connected outerplanar graph (with bridges or multiple
components), and its outer-face boundary closed walk serves as
$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices
multiple times.
In the special case $d = 0$ with single-vertex source $S = \{v_0\}$,
$R_{C'}$ is the star of $v_0$, a topological closed disk with one
boundary cycle (the link of $v_0$); the corresponding tire graph has
degenerate outer boundary $\{v_0\}$.
\end{remark}
\begin{proposition}[Edge--vertex coloring bijection for $D(T)$] \begin{proposition}[Edge--vertex coloring bijection for $D(T)$]
\label{prop:edge-vertex-bijection} \label{prop:edge-vertex-bijection}
papers/coloring_nested_tire_graphs/paper.aux
+12 -5
View File
@@ -6,6 +6,16 @@
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
\newlabel{fig:dual-depth}{{1}{2}} \newlabel{fig:dual-depth}{{1}{2}}
\newlabel{def:tire-graph}{{1.5}{2}} \newlabel{def:tire-graph}{{1.5}{2}}
\@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{rem:tire-counts}{{1.6}{3}}
\newlabel{prop:no-level-d-pinch}{{1.7}{3}}
\newlabel{lem:tire-component}{{1.8}{4}}
\citation{bauerfeld-depth}
\citation{bauerfeld-depth}
\newlabel{thm:tread-partition}{{1.9}{6}}
\newlabel{rem:tire-component-degenerate}{{1.10}{6}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}}
\bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-depth}{1}
\bibcite{bauerfeld-nested-tire-duals}{2} \bibcite{bauerfeld-nested-tire-duals}{2}
\newlabel{tocindent-1}{0pt} \newlabel{tocindent-1}{0pt}
@@ -13,8 +23,5 @@
\newlabel{tocindent1}{17.77782pt} \newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt} \newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt} \newlabel{tocindent3}{0pt}
\@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 } \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent }
\newlabel{fig:tire-example}{{2}{3}} \gdef \@abspage@last{7}
\newlabel{rem:tire-counts}{{1.6}{3}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
\gdef \@abspage@last{3}
papers/coloring_nested_tire_graphs/paper.log
+37 -31
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) 27 MAY 2026 01:01 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:13
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -192,47 +192,53 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
e e
)) ))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
<fig_dual_depth.png, id=21, 642.8015pt x 606.265pt> <fig_dual_depth.png, id=22, 642.8015pt x 606.265pt>
File: fig_dual_depth.png Graphic file (type png) File: fig_dual_depth.png Graphic file (type png)
<use fig_dual_depth.png> <use fig_dual_depth.png>
Package pdftex.def Info: fig_dual_depth.png used on input line 117. Package pdftex.def Info: fig_dual_depth.png used on input line 122.
(pdftex.def) Requested size: 251.9989pt x 237.67276pt. (pdftex.def) Requested size: 251.9989pt x 237.67276pt.
[2 <./fig_dual_depth.png>] [2 <./fig_dual_depth.png>]
<fig_tire_example.png, id=27, 559.64081pt x 375.804pt> <fig_tire_example.png, id=28, 559.64081pt x 375.804pt>
File: fig_tire_example.png Graphic file (type png) File: fig_tire_example.png Graphic file (type png)
<use fig_tire_example.png> <use fig_tire_example.png>
Package pdftex.def Info: fig_tire_example.png used on input line 172. Package pdftex.def Info: fig_tire_example.png used on input line 177.
(pdftex.def) Requested size: 280.79956pt x 188.56097pt. (pdftex.def) Requested size: 280.79956pt x 188.56097pt.
[3 <./fig_tire_example.png>] (./paper.aux) ) [3 <./fig_tire_example.png>] [4] [5] [6] [7]
(./paper.aux) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
3009 strings out of 478268 3017 strings out of 478268
42011 string characters out of 5846347 42169 string characters out of 5846347
337182 words of memory out of 5000000 344234 words of memory out of 5000000
21056 multiletter control sequences out of 15000+600000 21063 multiletter control sequences out of 15000+600000
475666 words of font info for 53 fonts, out of 8000000 for 9000 476364 words of font info for 55 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191 1302 hyphenation exceptions out of 8191
69i,7n,76p,625b,278s stack positions out of 10000i,1000n,20000p,200000b,200000s 69i,8n,76p,625b,284s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dis </usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
t/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist m/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/ m/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fo /cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
nts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/font cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
s/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/ mi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi
type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ 7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.
e1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/ pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf
public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb><
lic/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></us
ic/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/l
/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/loca
msfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local
fonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf /texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/t
onts/cm/cmti8.pfb> exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/tex
Output written on paper.pdf (3 pages, 457250 bytes). live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texli
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texliv
e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/
2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/texl
ive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
Output written on paper.pdf (7 pages, 539071 bytes).
PDF statistics: PDF statistics:
94 PDF objects out of 1000 (max. 8388607) 138 PDF objects out of 1000 (max. 8388607)
54 compressed objects within 1 object stream 82 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
+285 -9
View File
@@ -44,15 +44,20 @@
\dedicatory{} \dedicatory{}
\begin{abstract} \begin{abstract}
We establish the foundational definitions for studying the We establish the foundational structure of nested
Four Colour Theorem through nested level-structures on plane level-induced tire decompositions of a plane triangulation $G$.
triangulations. A \emph{level source} of a triangulation $G$ A \emph{level source} of $G$ induces a BFS layering of $G$ and
induces a BFS layering of $G$, which in turn endows the inner endows the inner planar dual $G'$ with a \emph{dual depth}
planar dual $G'$ with a \emph{dual depth} grading. We isolate the grading. The basic object of study is the \emph{tire graph}
basic object of study --- the \emph{tire graph} $T$, a plane graph $T$ --- a plane graph whose outer and inner boundaries bound a
whose outer and inner boundaries bound the \emph{tire tread} $R$, closed planar region, the \emph{tire tread} $R$, triangulated by
a closed region triangulated by the \emph{annular edges} the \emph{annular edges} $E_{\mathrm{ann}}$. Our main structural
$E_{\mathrm{ann}}$ --- and record its face/edge counts. result, the \emph{tire-component lemma}, exhibits each connected
component of $G'_d$ as a tire graph; the \emph{tire-tread
partition theorem} consequence shows the resulting tire treads
partition the bounded faces of $G$. Coloring questions on
$G$ thereby factor through coloring questions on the
individual treads.
\end{abstract} \end{abstract}
\maketitle \maketitle
@@ -190,6 +195,277 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
\end{remark} \end{remark}
\begin{proposition}[Source-side simple-cycle property]
\label{prop:no-level-d-pinch}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
single-vertex source $v_0$. Let $d \geq 1$, $v \in L_d$, and let
$C'$ be a connected component of $G'_d$ such that $v$ is incident to
some face in $F_{C'}$. Then the depth-$d$ faces in $F_{C'}$ incident
to $v$ form a single contiguous arc in $v$'s rotation in $\Pi_G$.
Equivalently: for any such component, the source-side boundary of
$R_{C'}$ is a simple cycle in $L_d$ (no cut-vertices at level $d$).
\end{proposition}
\begin{proof}
Suppose for contradiction that the depth-$d$ faces in $F_{C'}$ at $v$
lie in two or more disjoint arcs of $v$'s rotation. Adjacent vertices
in $G$ differ in level by at most $1$, so a face at $v$ has depth
exactly $d$ iff both other vertices have level $\geq d$, and depth
$\leq d-1$ iff at least one has level $d-1$. Hence the gaps between
the depth-$d$ arcs at $v$ are populated by level-$(d-1)$ neighbours of
$v$, occurring in at least two disjoint arcs of $v$'s rotation. Pick
$p$ in one such gap and $q$ in another.
The BFS ball $G[L_{<d}]$ is connected, so there exists a simple path
$P$ in $G[L_{<d}]$ from $p$ to $q$. Define the closed walk
\[
W \;:=\; v \to p \to P \to q \to v.
\]
Every vertex of $P$ lies in $L_{<d}$, while $\ell(v) = d$, so $v$ is
distinct from every vertex of $P$; $P$ is simple, so its internal
vertices are distinct; and $p \neq q$ since they lie in different gaps.
Hence $W$ is a simple cycle in $G$.
By the Jordan curve theorem, the planar embedding of $W$ divides $\Pi_G$
into two regions. In $v$'s rotation, the edges $v-p$ and $v-q$ lie at
two specific positions, and they split the rotation into two arcs;
each arc lies in one of the two regions determined by $W$. By choice
of $p, q$, the two arcs of depth-$d$ faces at $v$ in $F_{C'}$ lie in
different regions of $W$ (i.e., one arc on each side).
Since $C'$ is connected in $G'$ and contains depth-$d$ faces in both
arcs, there is a dual path $f_1, f_2, \ldots, f_k$ in $G'_d$ with
$f_1, f_k \in F_{C'}$ incident to $v$ in different arcs, and with the
intermediate faces $f_2, \ldots, f_{k-1}$ not incident to $v$ (a
shortest such dual path). Consecutive faces $f_i, f_{i+1}$ share an
edge $e_i$ of $G$; for $i \geq 2$, both endpoints of $e_i$ lie in
$L_{\geq d}$ (since neither $f_i$ nor $f_{i+1}$ is incident to $v$,
all six vertices of these two triangles lie in $L_{\geq d}$). In
particular, $e_i$ shares no endpoint with $W$ except possibly $v$ ---
and $v$ is excluded from $f_2, \ldots, f_{k-1}$.
A planar edge with neither endpoint on a simple closed planar curve
$W$ has both of its incident faces on the same side of $W$. Applying
this to each $e_i$ ($i \geq 2$) inductively: starting from $f_2$ on
the same side of $W$ as $f_1$ (their shared edge $e_1 = w-w'$ opposite
to $v$ in $f_1$ has $w, w' \in L_{\geq d}$ and hence is not on $W$),
the path $f_2 \to f_3 \to \cdots \to f_{k-1} \to f_k$ stays on one
side of $W$.
But $f_1$ and $f_k$ lie on different sides of $W$ (by construction),
contradicting the conclusion that the entire path lies on one side.
\end{proof}
\begin{lemma}[Tire-component lemma]
\label{lem:tire-component}
Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be a level
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
\[
G'_d \;:=\; G'\bigl[\,\{d_f \in V(G') : \delta_G(d_f) = d\}\,\bigr]
\]
be the inner-dual subgraph on dual vertices of dual depth $d$, and let
$C'$ be a connected component of $G'_d$. Write
$F_{C'} := \{f : d_f \in V(C')\}$,
$V_{C'} := \bigcup_{f \in F_{C'}} V(f)$, and let $C := G[V_{C'}]$ inherit
its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f
\subseteq |\Pi_G|$.
Then $C$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. Its outer boundary
$B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
boundary closed walk of $O$ in the inherited embedding (a simple cycle
when $O$ is $2$-connected, a non-simple closed walk in general). The
triangular faces of $C$ inside the closed boundary region are exactly
the faces of $G$ in $F_{C'}$.
\end{lemma}
\begin{proof}
\emph{Outerplanarity of the two level parts.} By construction $S$
lies on the outer face of $\Pi_G$, so the outerplanarity lemma of
\cite{bauerfeld-depth} applies directly with $(G, \Pi_G, S)$, giving
that $G[L_{d'}]$ is
outerplanar for each $d' \geq 0$. Subgraphs of outerplanar graphs are
outerplanar, so $G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ are
both outerplanar.
\emph{Layer containment.} Each $f \in F_{C'}$ has at least one vertex
at level $d$, and adjacent vertices in $G$ differ in level by at most
$1$; combined with $\delta_G(d_f) = d$, this forces
$V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup
L_{d+1}$, and $C$ has vertex partition
$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
\emph{Boundary edges are monochromatic in level.} Each edge $e$ on
$\partial R_{C'}$ separates a face $f \in F_{C'}$ from a face
$f' \notin F_{C'}$. Because $f$ and $f'$ share the edge $e$, their
dual vertices are adjacent in $G'$; if both had depth $d$ they would
lie in the same component of $G'_d$, contradicting $d_{f} \in C'$ and
$d_{f'} \notin C'$. Hence $\delta_G(d_{f'}) \neq d$; combined with
the bounded-step property of $\delta$ across $G'$-adjacent faces,
$\delta_G(d_{f'}) \in \{d-1, d+1\}$.
\begin{itemize}
\item If $\delta_G(d_{f'}) = d - 1$, the third vertex $w$ of
$f' = \{u, v, w\}$ (where $u, v$ are the endpoints of $e$) has
$\ell(w) = d - 1$. Each of $u, v$ has $\ell \in \{d, d+1\}$
(from $V(f) \subseteq L_d \cup L_{d+1}$) and is adjacent to $w$,
forcing $\ell(u), \ell(v) \in \{d-2, d-1, d\} \cap \{d, d+1\} =
\{d\}$.
\item If $\delta_G(d_{f'}) = d + 1$, then all three vertices of $f'$
lie in $L_{\geq d+1}$, so in particular $\ell(u) = \ell(v) =
d + 1$.
\end{itemize}
Each connected boundary component thus carries a single type at every
edge: any vertex on a boundary component has two boundary edges
incident to it (by R1, see below), both of the same type, so its
level is fixed. Therefore each boundary component of $\partial R_{C'}$
is monochromatic in level.
\emph{Boundary structure.} Each connected component of
$\partial R_{C'}$ traces a closed walk in $G$ that, by the
monochromaticity above, lies entirely in $L_d$ or entirely in
$L_{d+1}$. By Proposition~\ref{prop:no-level-d-pinch}, the depth-$d$
faces of $F_{C'}$ at any $v \in L_d \cap V_{C'}$ form a single
contiguous arc in $v$'s rotation, so the source-side boundary walk
visits each $L_d$-vertex of $V_{C'}$ exactly once: it is a simple
cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces
may split into multiple arcs of $v$'s rotation; this corresponds
exactly to $v$ being a cut-vertex of $O$, and the inner-side
boundary walk visits $v$ correspondingly many times --- which is
already accommodated by Definition~\ref{def:tire-graph} (where
$B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not
necessarily a simple cycle).
\emph{Outer boundary.} Because $S$ lies on the outer face of $\Pi_G$,
the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$
in the embedding. In the inherited embedding of $C$, the unique
unbounded face is the merged region containing the rest of $\Pi_G$
outside $R_{C'}$ on the $S$ side, so its boundary --- a simple cycle
on $L_d$ (or a single vertex when $V_{C'} \cap L_d = \{v_0\}$, the
$d = 0$ case) --- serves as $B_{\mathrm{out}}$. We set
$B_{\mathrm{out}} := G[V_{C'} \cap L_d]$ if this is a cycle, and
the single vertex $\{v_0\}$ in the degenerate case.
\emph{Inner outerplanar graph.} By the outerplanarity lemma of
\cite{bauerfeld-depth}, $G[V_{C'} \cap L_{d+1}]$ is outerplanar. We set $O :=
G[V_{C'} \cap L_{d+1}]$. The boundary curve(s) of $R_{C'}$ on the
$L_{d+1}$ side are exactly the boundary of $O$'s outer face in the
inherited embedding; this outer-face boundary is a single closed walk
that traces around $O$ from the outside, traversing any bridge edge
twice and visiting cut-vertices multiple times. This walk is the
inner boundary $B_{\mathrm{in}}$. No further restriction on $O$'s
internal structure is needed: when $R_{C'}$ has more than two
boundary components in the surface-classification sense (i.e.\
several disjoint simple cycles on $L_{d+1}$), these correspond
precisely to the multiple connected components or bridge crossings of
$O$, and the outer-face boundary closed walk of $O$ captures them
collectively.
\emph{Tire structure.} The triangular faces of $C$ inside the closed
boundary region are by construction the depth-$d$ faces in $F_{C'}$,
and the edges of $C$ are $E(B_{\mathrm{out}}) \cup E(O) \cup
E_{\mathrm{ann}}$ where $E_{\mathrm{ann}}$ are the edges of $G$
between $V_{C'} \cap L_d$ and $V_{C'} \cap L_{d+1}$ that bound a face
of $F_{C'}$.
\end{proof}
\begin{theorem}[Tire treads partition the bounded faces]
\label{thm:tread-partition}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and
let $S \subseteq V(G)$ be a level source lying on the outer face.
For each $d \ge 0$ and each connected component $C'$ of $G'_d$, let
$T^{(d, C')}$ denote the tire graph supplied by
Lemma~\ref{lem:tire-component}, with tire tread
$R_{C'} \subseteq |\Pi_G|$. Then the collection of treads
\[
\mathcal{R}(G, S) \;:=\;
\bigl\{\, R_{C'} \,:\, d \ge 0,\;
C' \text{ a connected component of } G'_d \,\bigr\}
\]
partitions the bounded part of $|\Pi_G|$:
\begin{enumerate}
\item[(i)] every bounded face $f$ of $G$ is contained in exactly
one tread $R_{C'} \in \mathcal{R}(G, S)$;
\item[(ii)] distinct treads in $\mathcal{R}(G, S)$ have disjoint
interiors and may share only boundary edges or vertices.
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{Existence and uniqueness.} Each bounded face $f \in F(G)$
has a uniquely-defined dual depth $\delta_G(d_f) \in \mathbb{Z}_{\ge
0}$, so the dual vertex $d_f$ lies in $G'_d$ for $d =
\delta_G(d_f)$ and in no other $G'_{d'}$. Within $G'_d$, the
vertex $d_f$ belongs to exactly one connected component $C'$. By
Lemma~\ref{lem:tire-component}, $F_{C'}$ is precisely the set of
faces $f' \in F(G)$ with $d_{f'} \in V(C')$; in particular $f \in
F_{C'}$, hence $f \subseteq R_{C'}$.
For any other tread $R_{C''} \in \mathcal{R}(G, S)$, the
component $C''$ is either at a different depth $d' \ne d$ (in
which case $F_{C''}$ consists of depth-$d'$ faces and $f \notin
F_{C''}$) or at depth $d$ but a different component $C'' \ne C'$
(in which case the two components are vertex-disjoint in $G'_d$,
so again $f \notin F_{C''}$). In both cases $f \notin R_{C''}$
(more precisely, $f$ is not one of the triangular faces of $G$ in
$F_{C''}$, so $f$'s interior is not contained in $R_{C''}$).
\emph{Disjoint interiors.} Each tread $R_{C'}$ is the union of
its triangular faces $F_{C'} \subseteq F(G)$; distinct treads
correspond to disjoint $F_{C'}$ (by the argument above), and the
interiors of distinct $G$-faces are disjoint. Hence interiors of
distinct treads are disjoint.
\emph{Coverage.} Conversely, every bounded $f \in F(G)$ has $d_f
\in V(G')$ with some dual depth $d$, and thus lies in $R_{C'}$
where $C'$ is its component of $G'_d$. So $\bigcup_{R \in
\mathcal{R}(G, S)} R$ contains every bounded face of $G$.
\end{proof}
\begin{remark}
\label{rem:tire-component-degenerate}
Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
degenerate. At $d = 0$ with single-vertex source $S = \{v_0\}$ the
unique component of $G'_0$ has $V_{C'} \cap L_0 = \{v_0\}$ as the
degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the
link of $v_0$ in $G$) as the inner boundary. Symmetrically, at
$d = D_{\max}$, $V_{C'} \cap L_{D_{\max}+1} = \emptyset$ degenerates
to a single deepest vertex serving as the \emph{inner} boundary, with
the level-$D_{\max}$ cycle as the outer boundary.
\end{remark}
\begin{remark}
\label{rem:tire-no-extra-hypotheses}
Two structural features of $R_{C'}$ that might at first appear to
obstruct the tire-graph conclusion are both already accommodated by
Definition~\ref{def:tire-graph}:
\emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may
have the faces of $F_{C'}$ incident to it split into two or more
arcs in $v$'s rotation in $\Pi_G$, separated by faces of higher
depth. This corresponds exactly to $v$ being a cut-vertex of
$O = G[V_{C'} \cap L_{d+1}]$, and the inner boundary closed walk
$B_{\mathrm{in}}$ then visits $v$ multiple times --- once for each
arc. No additional hypothesis is needed.
\emph{Multi-hole topology of $R_{C'}$.} Even when $R_{C'}$ encloses
several disjoint depth-$>d$ sub-regions, the inner outerplanar graph
$O$ captures the multi-hole structure as a disconnected or
non-$2$-connected outerplanar graph (with bridges or multiple
components), and its outer-face boundary closed walk serves as
$B_{\mathrm{in}}$ traversing bridges twice and visiting cut-vertices
multiple times.
In the special case $d = 0$ with single-vertex source $S = \{v_0\}$,
$R_{C'}$ is the star of $v_0$, a topological closed disk with one
boundary cycle (the link of $v_0$); the corresponding tire graph has
degenerate outer boundary $\{v_0\}$.
\end{remark}
\begin{thebibliography}{9} \begin{thebibliography}{9}