diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_internally_6_connected.py b/papers/coloring_nested_tire_graphs/experiments/draw_internally_6_connected.py
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index 0000000..a992b0e
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/experiments/draw_internally_6_connected.py
@@ -0,0 +1,116 @@
+"""Draw the icosahedron as the canonical "internally 6-connected"
+planar triangulation. Highlight a 5-vertex cut that isolates
+exactly one vertex (the only kind of small cut permitted by
+Birkhoff's condition).
+
+Internally 6-connected (Birkhoff 1913): a planar triangulation in
+which every vertex cut of size ≤ 5 either fails to separate, OR is
+a 5-cut whose two sides have one side equal to a single vertex.
+
+Equivalently: no separating 3- or 4-cycle (these are forbidden);
+no separating 5-cycle that isolates more than one vertex on either
+side.
+
+The icosahedron satisfies this: every vertex has degree 5, and the
+5 neighbors of any vertex form a 5-cycle whose removal isolates
+that vertex (1 on one side, 6 on the other).
+"""
+import os, sys
+import matplotlib.pyplot as plt
+from sage.all import graphs
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+
+
+def main():
+    G = graphs.IcosahedralGraph()
+    G.is_planar(set_embedding=True)
+    pos = G.layout(layout='planar')
+
+    # Pick a "center" vertex (one with small distance from center of layout)
+    center_v = 0  # arbitrary
+    neighbors = list(G.neighbors(center_v))
+    print(f'Vertex {center_v} has {len(neighbors)} neighbors: {neighbors}')
+
+    # Verify they form a 5-cycle in the induced subgraph
+    nbr_graph = G.subgraph(neighbors)
+    print(f'Induced subgraph on neighbors: {nbr_graph.size()} edges, '
+          f'is_cycle={nbr_graph.is_cycle()}')
+
+    # Removing 5 neighbors: what's left
+    G_minus = G.copy()
+    G_minus.delete_vertices(neighbors)
+    comps = G_minus.connected_components()
+    print(f'After removing the 5 neighbors: {len(comps)} components, '
+          f'sizes={sorted(len(c) for c in comps)}')
+    # Expect: one component of size 1 (just center_v), another of size 6.
+
+    fig, axes = plt.subplots(1, 2, figsize=(13, 6))
+
+    # LEFT: the icosahedron with the cut highlighted
+    ax = axes[0]
+    for u, v in G.edges(labels=False):
+        x1, y1 = pos[u]; x2, y2 = pos[v]
+        if u == center_v or v == center_v:
+            color = '#CCCCCC'; lw = 1.0
+        elif u in neighbors and v in neighbors:
+            # Edge of the 5-cycle separator
+            color = '#1166CC'; lw = 4.0
+        else:
+            color = '#888888'; lw = 1.2
+        ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=2)
+    for v in G.vertices():
+        x, y = pos[v]
+        if v == center_v:
+            ax.plot(x, y, 'o', color='#DD2222', markersize=14, zorder=4,
+                    markeredgecolor='black', markeredgewidth=1.5)
+            ax.annotate(f'{v}\n(isolated)', (x, y),
+                        textcoords='offset points', xytext=(0, 14),
+                        ha='center', fontsize=9, color='#AA0000')
+        elif v in neighbors:
+            ax.plot(x, y, 'o', color='#1166CC', markersize=11, zorder=3,
+                    markeredgecolor='black', markeredgewidth=1.2)
+            ax.annotate(str(v), (x, y),
+                        textcoords='offset points', xytext=(8, 8),
+                        fontsize=8, color='#0044AA')
+        else:
+            ax.plot(x, y, 'o', color='#444444', markersize=8, zorder=2)
+            ax.annotate(str(v), (x, y),
+                        textcoords='offset points', xytext=(8, 8),
+                        fontsize=8, color='gray')
+    ax.set_aspect('equal'); ax.axis('off')
+    ax.set_title(f'Icosahedron: 5-vertex cut (blue) isolating one vertex (red).\n'
+                 f'This is the ONLY kind of small cut permitted '
+                 f'by Birkhoff\'s condition.', fontsize=10)
+
+    # RIGHT: schematic showing 4 cases (3 forbidden, 1 allowed)
+    ax = axes[1]
+    ax.set_xlim(0, 4); ax.set_ylim(0, 4)
+    ax.set_aspect('equal'); ax.axis('off')
+    cases = [
+        ('3-vertex cut\n(separating triangle)', 'FORBIDDEN', 'red', 3.6, 3),
+        ('4-vertex cut\n(separating quadrilateral)', 'FORBIDDEN', 'red', 3.6, 2),
+        ('5-cut isolating ≥ 2 vertices', 'FORBIDDEN', 'red', 3.6, 1),
+        ('5-cut isolating exactly 1 vertex', 'ALLOWED', 'green', 3.6, 0),
+    ]
+    ax.text(2.0, 3.7, 'Birkhoff (1913): internally 6-connected',
+            fontsize=11, weight='bold', ha='center')
+    for i, (label, status, color, y_label, y_pos) in enumerate(cases):
+        y = 3.0 - i * 0.7
+        ax.text(0.1, y, label, fontsize=9, va='center')
+        ax.text(2.8, y, status, fontsize=10, weight='bold', color=color,
+                va='center')
+    ax.text(2.0, -0.4,
+            'A minimum 4CT counterexample, if one exists,\n'
+            'must satisfy all of the above.',
+            fontsize=9, ha='center', style='italic')
+
+    plt.tight_layout()
+    out = os.path.join(os.path.dirname(HERE), 'notes',
+                       'icosahedron_internally_6_connected.pdf')
+    fig.savefig(out, bbox_inches='tight', dpi=120)
+    print(f'\nWrote {out}')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux
new file mode 100644
index 0000000..a79beaf
--- /dev/null
+++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux
@@ -0,0 +1,2 @@
+\relax 
+\gdef \@abspage@last{3}
diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log
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+++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log
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diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.pdf b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.pdf
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+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{graphicx}
+\usepackage{geometry}
+\usepackage{booktabs}
+\usepackage{tikz}
+\usetikzlibrary{backgrounds}
+\geometry{margin=1in}
+
+\title{Birkhoff's condition: minimum 4CT counterexamples are
+internally $6$-connected}
+\author{}
+\date{}
+
+\newtheorem*{defn}{Definition}
+\newtheorem*{thm}{Theorem}
+\newtheorem*{lem}{Lemma}
+\newtheorem*{rem}{Remark}
+
+\begin{document}
+\maketitle
+
+\section*{The condition}
+
+\begin{defn}[Internally $6$-connected]
+A planar triangulation $G$ is \emph{internally $6$-connected} if
+every vertex cut of $G$ of size $\le 5$ either fails to separate
+$G$, or is a $5$-cut whose two sides have one side equal to a
+single vertex.
+
+Equivalently: $G$ has no separating triangle (3-cycle), no
+separating quadrilateral (4-cycle), and any separating pentagon
+(5-cycle) isolates exactly one vertex on one side.
+\end{defn}
+
+\begin{thm}[Birkhoff 1913]
+If a minimum $4$-colour counterexample (= minimum planar
+triangulation requiring more than $4$ colours for a proper vertex
+colouring) exists, it must be internally $6$-connected.
+\end{thm}
+
+The idea behind the proof: if $G$ has a separating $k$-cycle for
+$k \in \{3, 4, 5\}$ with $\ge 2$ vertices on each side, one can
+$4$-colour each side independently and patch the colourings along
+the cycle, contradicting minimality.  Only the $5$-cycle isolating
+$1$ vertex is irreducible by this argument.
+
+\section*{Cases at a glance}
+
+\begin{center}
+\begin{tikzpicture}[scale=1.1]
+  \node[anchor=west] at (0, 4.3) {\textbf{Internally $6$-connected $=$ all four below:}};
+  % case row
+  \node[anchor=west] at (0, 3.6) {\textbullet\, no separating $3$-cycle};
+  \node[red, anchor=west] at (5.0, 3.6) {FORBIDDEN};
+  \node[anchor=west] at (0, 2.9) {\textbullet\, no separating $4$-cycle};
+  \node[red, anchor=west] at (5.0, 2.9) {FORBIDDEN};
+  \node[anchor=west] at (0, 2.2) {\textbullet\, no separating $5$-cycle isolating $\ge 2$ on either side};
+  \node[red, anchor=west] at (5.0, 2.2) {FORBIDDEN};
+  \node[anchor=west] at (0, 1.5) {\textbullet\, separating $5$-cycle isolating $1$ vertex};
+  \node[green!50!black, anchor=west] at (5.0, 1.5) {ALLOWED};
+  % Schematic: a 5-cycle with 1 isolated vertex on one side
+  \begin{scope}[shift={(7, 2.7)}, scale=0.5]
+    \node[circle, fill=red!80!black, inner sep=2pt] (c) at (0, 0) {};
+    \foreach \i in {0,...,4} {
+      \pgfmathsetmacro{\ang}{72*\i+90}
+      \node[circle, fill=blue!80!black, inner sep=1.8pt] (n\i) at (\ang:1.4) {};
+      \draw[blue, thick] (c) -- (n\i);
+    }
+    \foreach \i in {0,...,4} {
+      \pgfmathtruncatemacro{\j}{mod(\i+1,5)}
+      \draw[blue, very thick] (n\i) -- (n\j);
+    }
+    \node[red] at (0, -1.95) {\small allowed};
+  \end{scope}
+\end{tikzpicture}
+\end{center}
+
+The right-hand schematic shows the only permitted case: a vertex
+(red) surrounded by its $5$ neighbours (blue) which form a $5$-cycle.
+Removing the $5$ blue vertices isolates the red one.  This is what
+the next section's picture shows in the icosahedron.
+
+\section*{Icosahedron: the smallest example}
+
+The icosahedral graph (the $1$-skeleton of the regular icosahedron)
+has $12$ vertices, all of degree $5$, and is a planar triangulation.
+Every vertex's $5$ neighbours form a $5$-cycle (the ``link'' of the
+vertex); removing them isolates the vertex.  This is the only kind
+of small cut: the icosahedron is internally $6$-connected.
+
+\begin{center}
+\includegraphics[width=\textwidth]{icosahedron_internally_6_connected.pdf}
+\end{center}
+
+On the left, the icosahedron's planar (Schlegel) embedding: the red
+vertex (here vertex $0$) is removed from the graph by the $5$-cut
+consisting of its $5$ blue neighbours $\{1, 5, 7, 8, 11\}$.  The
+blue $5$-cycle is the separator; on one side is the isolated red
+vertex, on the other are the remaining $6$ vertices.  This is the
+allowed Birkhoff configuration.
+
+\section*{Why this matters for the framework}
+
+For our cut-tire chain DP framework, we test on graphs whose primal
+triangulation is internally $6$-connected:
+\begin{itemize}
+\item \textbf{Icosahedron} ($V = 12$): smallest example.  Its
+      cubic dual = dodecahedron ($V = 20$).
+\item \textbf{Pentakis dodecahedron} ($V = 32$): icosahedron with
+      each face subdivided.  Cubic dual = truncated icosahedron =
+      ``Buckminsterfullerene'' ($V = 60$).
+\item \textbf{Holton--McKay graphs} ($V = 21$ primal triangulation,
+      $V = 38$ cubic dual): the smallest non-Hamiltonian
+      internally-$6$-connected cubic plane graphs, candidates for
+      a minimum counterexample in Tait's reduction even though all
+      are $3$-edge-colourable.
+\end{itemize}
+
+If our framework's claims hold for all internally $6$-connected
+triangulations, then a minimum $4$CT counterexample (if it existed)
+would be in the framework's domain --- and any structural obstruction
+the framework finds would refute the counterexample's existence.
+
+\section*{Sanity check}
+
+Sage verification (in
+\texttt{experiments/draw\_internally\_6\_connected.py}):
+\begin{verbatim}
+Vertex 0 has 5 neighbors: [1, 5, 7, 8, 11]
+Induced subgraph on neighbors: 5 edges, is_cycle=True
+After removing the 5 neighbors: 2 components, sizes=[1, 6]
+\end{verbatim}
+
+So vertex $0$'s neighbourhood is indeed a $5$-cycle, removing it
+isolates vertex $0$ from the other $6$ vertices.  By symmetry the
+same holds at every vertex of the icosahedron.
+
+\end{document}
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