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coloring_nested_tire_graphs: new note on Birkhoff's internally-6-connected condition

Browse Source 
NEW NOTE: birkhoff_internally_6_connected.tex (3 pages)
NEW SCRIPT: experiments/draw_internally_6_connected.py
NEW FIGURE: icosahedron_internally_6_connected.pdf

States and illustrates the Birkhoff (1913) condition that any
minimum 4CT counterexample must be internally 6-connected:

  - No separating 3-cycle.
  - No separating 4-cycle.
  - No separating 5-cycle isolating ≥ 2 vertices on either side.
  - Only separating 5-cycles isolating exactly 1 vertex.

The icosahedron is the canonical example: 12 vertices all of
degree 5; the 5 neighbors of every vertex form a 5-cycle whose
removal isolates that vertex.  Sage verification confirms this:

  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]

Note also lists the graphs used in our framework testing:
  - Icosahedron (12 v, dual = dodecahedron)
  - Pentakis dodecahedron (32 v, dual = Buckyball)
  - Holton-McKay graphs (21 v primal, 38 v dual)

All are internally 6-connected, hence in the framework's intended
domain.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 00:00:46 -04:00
parent 5e9e8ed09b
commit 95d020b113
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papers/coloring_nested_tire_graphs/experiments/draw_internally_6_connected.py
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"""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()
papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux
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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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