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 8ec54d9..70b9eb2 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png and b/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 3a21ced..b9a54c6 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -28,7 +28,7 @@ \newlabel{tocindent3}{0pt} \newlabel{rem:edge-vertex-corollary}{{1.14}{9}} \newlabel{def:tire-annular-subgraph}{{1.15}{9}} -\newlabel{def:partial-tire-facial-dual}{{1.16}{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 } 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). PDF statistics: 128 PDF objects out of 1000 (max. 8388607) 73 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 243654e..9aa98c8 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 9d1f540..3259206 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -587,14 +587,14 @@ the annulus, remains a planar embedding of $T'_{\mathrm{ann}}$ in the sense of $\Pi_G$). \end{definition} -\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)