diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 342465f..0b45a28 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) 29 MAY 2026 23:25 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 1 JUN 2026 00:17 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -546,7 +546,7 @@ mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m Here is how much of TeX's memory you used: 14062 strings out of 478268 279636 string characters out of 5846347 - 565983 words of memory out of 5000000 + 566981 words of memory out of 5000000 31885 multiletter control sequences out of 15000+600000 478218 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 @@ -576,7 +576,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb> -Output written on paper.pdf (16 pages, 929392 bytes). +Output written on paper.pdf (16 pages, 929386 bytes). PDF statistics: 192 PDF objects out of 1000 (max. 8388607) 116 compressed objects within 2 object streams diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index e785cfd..89a1c47 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 e1784c5..f2d6f64 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -298,17 +298,17 @@ outer face (such an embedding exists for any planar graph and any single-vertex source). For $d \geq 0$, let $C'$ be a connected component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$, bounding vertices $V_{C'}$, and region $R_{C'}$ as in -Definition~\ref{def:dual-component}; let $C := G[V_{C'}]$ inherit its +Definition~\ref{def:dual-component}; let $T_{C'} := G[V_{C'}]$ inherit its embedding from $\Pi_G$. -Then $C$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}. Its outer boundary +Then $T_{C'}$, 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 +triangular faces of $T_{C'}$ inside the closed boundary region are exactly the faces of $G$ in $F_{C'}$. \end{lemma} @@ -325,7 +325,7 @@ both outerplanar. 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 +L_{d+1}$, and $T_{C'}$ 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 @@ -370,7 +370,7 @@ 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 +in the embedding. In the inherited embedding of $T_{C'}$, 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 @@ -393,9 +393,9 @@ 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 +\emph{Tire structure.} The triangular faces of $T_{C'}$ 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 +and the edges of $T_{C'}$ 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 $F_{C'}$. @@ -456,7 +456,7 @@ where $C'$ is its component of $G'_d$. So $\bigcup_{R \in \begin{remark} \label{rem:tire-component-degenerate} -Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be +Either boundary part of $T_{C'}$ 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