coloring_nested_tire_graphs: extend Def 1.5 with degenerate boundaries; add tire-component lemma + example figure

- Extends Definition 1.5 (tire graph) to allow either B_out or B_in to
  be a single vertex (a "degenerate" boundary); the tire is then a
  triangulated disk with that vertex as apex.
- Adds Remark 1.6 with vertex/edge/triangle counts for both the
  non-degenerate (annulus) and degenerate (disk) cases.
- Adds Lemma 1.7 (tire-component lemma): for a maximal planar graph G
  with level source S and depth d ≥ 0, every connected component C' of
  the dual subgraph on dual vertices of depth d induces a subgraph C
  on G that is a tire graph; its two boundary parts are the level-d
  and level-(d+1) induced subgraphs (in either order), and its
  triangular faces are exactly the faces of G in F_{C'}.
- Adds Remark 1.8 noting the d=0 degenerate-source case as an example.
- Adds a figure (fig_tire_example.png) illustrating the definition with
  m=6 outer cycle, k=4 inner cycle, one chord in O, plus a legend
  identifying B_out, B_in, O's chord, and E_ann.
- Adds experiments/tire_def_figure.py to regenerate the figure.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/coloring_nested_tire_graphs/experiments/tire_def_figure.py

@@ -0,0 +1,113 @@
+"""Generate the example figure for Definition 1.5 (tire graph) of the
+paper.  Produces fig_tire_example.png at the paper's top level.
+
+Picks a specific small tire (m=6 outer cycle, k=4 inner cycle with one
+chord) so that all four named parts B_out, O, B_in, E_ann are visible
+and the annular triangles are individually legible.
+"""
+import math
+import os
+import sys
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+
+from tire_graph import random_tire, planar_positions
+
+
+def draw_tire_def(tire, filename):
+    m, k = tire['m'], tire['k']
+    outer, inner = tire['outer'], tire['inner']
+    edges = tire['edges']
+
+    R_out, R_in = 1.0, 0.45
+    pos = planar_positions(tire, R_out=R_out, R_in=R_in)
+
+    fig, ax = plt.subplots(figsize=(6.5, 6.5))
+
+    # guide circles for reference
+    for r in (R_out + 0.04, R_in - 0.04):
+        ax.add_patch(patches.Circle((0, 0), r, fill=False,
+                                    edgecolor='lightgray',
+                                    linewidth=0.5, linestyle='--'))
+
+    outer_set = set(outer)
+    inner_set = set(inner)
+    C = {
+        'outer_cycle': '#1f77b4',
+        'inner_cycle': '#d62728',
+        'inner_chord': '#ff7f0e',
+        'spoke':       '#7f7f7f',
+    }
+
+    # classify and draw edges
+    for (u, v) in edges:
+        if u in outer_set and v in outer_set:
+            color, lw, label = C['outer_cycle'], 2.6, 'B_out'
+        elif u in inner_set and v in inner_set:
+            ia, ib = u - m, v - m
+            d = abs(ia - ib)
+            d = min(d, k - d)
+            if d == 1:
+                color, lw, label = C['inner_cycle'], 2.6, 'B_in'
+            else:
+                color, lw, label = C['inner_chord'], 1.8, 'O chord'
+        else:
+            color, lw, label = C['spoke'], 1.1, 'E_ann'
+        x1, y1 = pos[u]; x2, y2 = pos[v]
+        ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
+
+    # vertices
+    for v in outer:
+        x, y = pos[v]
+        ax.plot(x, y, 'o', color=C['outer_cycle'], markersize=14, zorder=2)
+        ax.annotate(str(v), (x, y), color='white', ha='center',
+                    va='center', fontsize=9, fontweight='bold', zorder=3)
+    for v in inner:
+        x, y = pos[v]
+        ax.plot(x, y, 'o', color=C['inner_cycle'], markersize=13, zorder=2)
+        ax.annotate(str(v), (x, y), color='white', ha='center',
+                    va='center', fontsize=8, fontweight='bold', zorder=3)
+
+    # legend
+    legend_items = [
+        plt.Line2D([], [], color=C['outer_cycle'], linewidth=2.6,
+                   label=r'$B_{\mathrm{out}}$ (outer boundary, $m=6$)'),
+        plt.Line2D([], [], color=C['inner_cycle'], linewidth=2.6,
+                   label=r'$B_{\mathrm{in}}$ (inner boundary, $k=4$)'),
+        plt.Line2D([], [], color=C['inner_chord'], linewidth=1.8,
+                   label=r'chord of $O$'),
+        plt.Line2D([], [], color=C['spoke'], linewidth=1.1,
+                   label=r'$E_{\mathrm{ann}}$ (annular edges)'),
+    ]
+    ax.legend(handles=legend_items, loc='upper left',
+              bbox_to_anchor=(1.0, 1.0), fontsize=10, frameon=False)
+
+    ax.set_xlim(-1.20, 1.20)
+    ax.set_ylim(-1.20, 1.20)
+    ax.set_aspect('equal')
+    ax.axis('off')
+
+    plt.savefig(filename, dpi=160, bbox_inches='tight')
+    plt.close()
+
+
+def main():
+    # m=6 outer, k=4 inner, 1 chord — try a few seeds to pick a clean one
+    paper_dir = os.path.abspath(os.path.join(HERE, '..'))
+    candidates = [(6, 4, 1, s) for s in (3, 5, 8, 13, 21)]
+    # Use seed=3 as the chosen example (good lattice-path balance)
+    for (m, k, nc, seed) in candidates[:1]:
+        tire = random_tire(m, k, n_chords=nc, seed=seed)
+        fn = os.path.join(paper_dir, 'fig_tire_example.png')
+        draw_tire_def(tire, fn)
+        print(f"  wrote {fn}")
+        print(f"    m={m}, k={k}, chords={tire['inner_chords']}, "
+              f"path={tire['lattice_path']}")
+
+
+if __name__ == '__main__':
+    main()

papers/coloring_nested_tire_graphs/paper.aux

@@ -1,12 +1,17 @@
 \relax 
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
 \newlabel{def:dual-depth}{{1.4}{1}}
+\@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{def:tire-graph}{{1.5}{2}}
+\newlabel{rem:tire-counts}{{1.6}{2}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{0pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@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 {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:dual-depth}{{1}{2}}
+\newlabel{fig:tire-example}{{2}{3}}
-\newlabel{def:tire-graph}{{1.5}{2}}
+\newlabel{lem:tire-component}{{1.7}{3}}
+\newlabel{rem:tire-component-degenerate}{{1.8}{3}}
 \gdef \@abspage@last{3}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -116,39 +116,94 @@ vertex.}
 
 \begin{definition}[Tire graph]
 \label{def:tire-graph}
-Let $C_{\mathrm{out}}$ be a simple cycle of length $m \geq 3$, and let
+A \emph{tire graph} consists of a plane graph $T$ together with two
-$O$ be an outerplanar graph whose outer-face boundary $C_{\mathrm{in}}$
+\emph{boundary parts} $B_{\mathrm{out}}, B_{\mathrm{in}} \subseteq T$
-is a simple cycle of length $k \geq 3$, with $V(C_{\mathrm{out}}) \cap
+and an \emph{inner outerplanar graph} $O \subseteq T$, where each of
-V(O) = \emptyset$.  A \emph{tire graph} on $(C_{\mathrm{out}}, O)$ is a
+$B_{\mathrm{out}}$ and the outer-face boundary $B_{\mathrm{in}}$ of $O$
-plane graph $T$ with
+is either
+\begin{itemize}
+  \item a simple cycle of length $\geq 3$, or
+  \item a single vertex (a \emph{degenerate} boundary),
+\end{itemize}
+with at most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ degenerate, and
+$V(B_{\mathrm{out}}) \cap V(O) = \emptyset$.  The vertex and edge sets
+of $T$ are
 \[
-    V(T) = V(C_{\mathrm{out}}) \cup V(O),
+    V(T) = V(B_{\mathrm{out}}) \cup V(O),
     \qquad
-    E(T) = E(C_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
+    E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
 \]
-where $E_{\mathrm{ann}}$ is a set of edges --- the \emph{annular edges}
+where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
---- such that, in the plane embedding of $T$, the closed annulus with
+property that, in the plane embedding of $T$, the closed planar region
-outer boundary $C_{\mathrm{out}}$ and inner boundary $C_{\mathrm{in}}$
+$R$ bounded externally by $B_{\mathrm{out}}$ and internally by
-is partitioned into triangular faces.  Equivalently, the bounded faces
+$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose
-of $T$ that are not faces of $O$ are all triangles, and together they
+union is $R$.  The region $R$ is a closed annulus when both
-tile the annular region between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$.
+$B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ are cycles, and a closed disk
+when exactly one of them is a single vertex.
 
-We call $C_{\mathrm{out}}$ the \emph{outer cycle}, $O$ the \emph{inner
+We call $B_{\mathrm{out}}$ the \emph{outer boundary}, $O$ the
-outerplanar graph}, and $C_{\mathrm{in}}$ the \emph{inner cycle} of
+\emph{inner outerplanar graph}, and $B_{\mathrm{in}}$ the \emph{inner
-$T$.  When $O = C_{\mathrm{in}}$ (the inner outerplanar graph has no
+boundary} of $T$.  A tire graph in which $B_{\mathrm{out}}$
-chords), $T$ is a tire graph \emph{with empty inner}; in general $O$
+(respectively $B_{\mathrm{in}}$) is a single vertex is said to have a
-contributes only chords inside the disk bounded by $C_{\mathrm{in}}$
+\emph{degenerate outer (respectively inner) boundary}; in either case
-and does not interact with $E_{\mathrm{ann}}$.
+$T$ is a triangulated closed disk with that vertex as apex.
 \end{definition}
 
+\begin{figure}[h]
+\centering
+\includegraphics[width=0.78\textwidth]{fig_tire_example.png}
+\caption{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.}
+\label{fig:tire-example}
+\end{figure}
+
 \begin{remark}
-A tire graph on $(C_{\mathrm{out}}, O)$ has $|V(C_{\mathrm{out}})| +
+\label{rem:tire-counts}
-|V(O)| = m + k$ vertices, exactly $m + k$ annular triangles
+Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$.  By
-in the annulus between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$ (by
+Euler's formula on the annular (resp.\ disk) region $R$, the tire graph
-Euler's formula on the annulus), and exactly $m + k$ annular edges
+has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
-in $E_{\mathrm{ann}}$, of which the $m + k$ triangles share their
+annular edges when neither boundary is degenerate; when exactly one
-three edges with the boundaries $E(C_{\mathrm{out}}) \cup
+boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
-E(C_{\mathrm{in}})$ and with each other.
+triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
+\end{remark}
+
+\begin{lemma}[Tire-component lemma]
+\label{lem:tire-component}
+Let $G$ be a maximal planar graph with fixed embedding $\Pi_G$ and let
+$S \subseteq V(G)$ be a level 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
+$C := G[V_{C'}]$ with the embedding inherited from $\Pi_G$.
+
+Then $C$, together with its inherited embedding, is a tire graph in the
+sense of Definition~\ref{def:tire-graph}: the two boundary parts
+$\{B_{\mathrm{out}}, B_{\mathrm{in}}\}$ of $C$ are the level-$d$
+subgraph $G[V_{C'} \cap L_d]$ and the level-$(d+1)$ subgraph
+$G[V_{C'} \cap L_{d+1}]$, in either order, and the triangular faces of
+$C$ inside its closed boundary region are exactly the faces of $G$ in
+$F_{C'}$.
+\end{lemma}
+
+\begin{remark}
+\label{rem:tire-component-degenerate}
+Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
+degenerate.  For instance, at $d = 0$ with single-vertex source
+$S = \{v_0\}$ the unique component of $G'_0$ has
+$V_{C'} \cap L_0 = \{v_0\}$ --- the degenerate boundary --- and
+$V_{C'} \cap L_1$ a cycle (the link of $v_0$ in $G$).  Which of the two
+parts is $B_{\mathrm{out}}$ and which is $B_{\mathrm{in}}$ depends on
+the orientation of the inherited embedding (equivalently, on which side
+of $C$ contains the rest of $\Pi_G$).
 \end{remark}
 
 \end{document}