From 65eeb607e102c025151130407460666cfc4f6898 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 23:06:34 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20rename=20"?= =?UTF-8?q?partial=20tire=20facial=20dual"=20=E2=86=92=20"tire=20annular?= =?UTF-8?q?=20face=20connector"?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Renames Definition 1.16 from "Partial tire facial dual" to "Tire annular face connector" to match the family of "tire annular ..." terminology (cf. Definition 1.15 "Tire annular subgraph"). Symbol T'_{f'} unchanged. Updates: - Definition 1.16 title and body wording - Label changed to def:tire-annular-face-connector - Figure suptitle in fig_facial_dual_choices.png - Module docstring of draw_facial_dual_choices.py Paper stays at 10 pages. Co-Authored-By: Claude Opus 4.7 --- .../experiments/draw_facial_dual_choices.py | 4 ++-- .../notes/fig_facial_dual_choices.png | Bin 75719 -> 75684 bytes papers/coloring_nested_tire_graphs/paper.aux | 2 +- papers/coloring_nested_tire_graphs/paper.log | 6 +++--- papers/coloring_nested_tire_graphs/paper.pdf | Bin 813353 -> 813184 bytes papers/coloring_nested_tire_graphs/paper.tex | 8 ++++---- 6 files changed, 10 insertions(+), 10 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py index 844e44c..ff7c7e2 100644 --- a/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py +++ b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py @@ -1,5 +1,5 @@ """For the bridge case where T'_ann = theta(1, p, q) has three faces, -illustrate how the partial tire facial dual T'_{f'} (Definition 1.15) +illustrate how the tire annular face connector T'_{f'} (Definition 1.16) depends on the choice of face f'. We use theta(1, 3, 3) = C_6 + chord (v_0, v_3) as the smallest @@ -155,7 +155,7 @@ def main(): show_external_for=ext_candidates, face_shade=shadeC) - fig.suptitle(r"Partial tire facial dual $T'_{f'}$ for the bridge case " + + fig.suptitle(r"Tire annular face connector $T'_{f'}$ for the bridge case " + r"($T'_{\mathrm{ann}} = \theta(1,3,3)$, three faces $A,B,C$)" + "\n" + r"Blue: edges of $T'_{f'}$. Dark circles: $V(f')$. " + r"Red squares: external $G'$-neighbors $u_v$ included via $v \in V(f')$.", diff --git a/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png b/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png index 8ec54d96e52625fdcb595559427dcb174d369c1d..70b9eb2bf3540881c4a7e01dfeebb29ffbb2c0f9 100644 GIT binary patch literal 75684 zcmce;g;&$<|35zUM!iu)MMObD5ou5)BnA=^($Y0lx}{?u=nwQE)X=YNQJcsYogQ53OYvwX);Nuxy6fZ7vL0qu@1BmSv^jxy;AlOQFKE^2UZD8m zxE~b_gZGkauGW^fY}{`qs=j4SD5$s>IGU>w*RxoV>FzvM>U2Lrz|OqJ@exPh?{n4u z#Ob?a3xw~^OnB z`tHo0-rSjuN!{JVr{>&yow77qf@L`?YIL2SPsXk*{!MQS3>m}|oE~zpi?23>j1(Jn z`w*v*cQZ2sA7vg>+@%mvy=yaV+ST^GJ!TH+e?DYyYY3%>*~IdZ&YhXRHV4%>li#vQ zv9^2mq)B`CxYbYD_etZ^UXoH>$3J}u-st3=Uu`DV?UqM9^v4D7ueY}zCUS=E4^9T+ z)AJ1qbm|^iy`pE3BEp;vmMnw<4U-6&xwEq@yj%0a1p#pU6^D5`m`vG43N7*_iPaKkf;4Q(7S@4b2*s4M={@~tkhe7vl|(z z1y@D#Q*OB2=To)O914gQD)|Iw@I!ber1LL5=6z3ijcrco!7lHw+Ock5<0C@Jvi{{% z4!R{~Q^5|r18f|h{FLJTCHMXpD4g^`cD14Hf{5jve%R}c9QlyF@vi7F2DX$(ZDGlA 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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 } diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 327433f..baa2443 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 22:49 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -244,7 +244,7 @@ LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 3041 strings out of 478268 - 43054 string characters out of 5846347 + 43057 string characters out of 5846347 344281 words of memory out of 5000000 21084 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 @@ -266,7 +266,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (10 pages, 813353 bytes). +Output written on paper.pdf (10 pages, 813184 bytes). 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-\begin{definition}[Partial tire facial dual] -\label{def:partial-tire-facial-dual} +\begin{definition}[Tire annular face connector] +\label{def:tire-annular-face-connector} With $G, G', T$ as in Definition~\ref{def:tire-annular-subgraph}, let $f'$ be a face of the tire annular subgraph $T'_{\mathrm{ann}}$ in its inherited embedding, and let $V(f') \subseteq V(T'_{\mathrm{ann}})$ denote the set of -vertices on the boundary walk of $f'$. The \emph{partial tire facial -dual at $f'$} is the subgraph +vertices on the boundary walk of $f'$. The \emph{tire annular face +connector at $f'$} is the subgraph \[ T'_{f'} \;:=\; \bigl(\,V(f') \cup N_{G'}(V(f'))\,,\; \{\,e \in E(G') : e \text{ is incident to } V(f')\,\}\,\bigr)