From 95d020b11379bc16708ab8b503ca76ad15c539a0 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 00:00:46 -0400 Subject: [PATCH] coloring_nested_tire_graphs: new note on Birkhoff's internally-6-connected condition MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../draw_internally_6_connected.py | 116 ++++ .../notes/birkhoff_internally_6_connected.aux | 2 + .../notes/birkhoff_internally_6_connected.log | 601 ++++++++++++++++++ .../notes/birkhoff_internally_6_connected.pdf | Bin 0 -> 161512 bytes .../notes/birkhoff_internally_6_connected.tex | 139 ++++ .../icosahedron_internally_6_connected.pdf | Bin 0 -> 39873 bytes 6 files changed, 858 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_internally_6_connected.py create mode 100644 papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log create mode 100644 papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.tex create mode 100644 papers/coloring_nested_tire_graphs/notes/icosahedron_internally_6_connected.pdf 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 new file mode 100644 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 new file mode 100644 index 0000000..eef8819 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log @@ -0,0 +1,601 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded 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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} diff --git a/papers/coloring_nested_tire_graphs/notes/icosahedron_internally_6_connected.pdf b/papers/coloring_nested_tire_graphs/notes/icosahedron_internally_6_connected.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1fb17adecb9c73014d1ddd0ed9e76c510ba389c7 GIT binary patch literal 39873 zcmd431yohd`!9|l-60|p2N94yhXaR@lZx2Y>+qr+!q&Aj9gsIoa{h6@SixG>Xt^PR(2L3*w0!g zdlPjt7mzN{t%MXnikZ6$h*Q=UAmPr>T>NJ)3)24^Igo!308TfYdM7i0x(o6-)y$mj zU7bvT@xbW^`d2hFwKBS6?+)|`2LAY79Ly*aUuGWaKOFs068@!Sha-=~OJ(ke zW35Qu-G1-4vZQbp3HeyG^@Y$Rz_K9_m;1 zSc3XgXC87xJqDyi%2#52z(J8OM6h3;8fLwCoH0gF_A zSTht(AFA&RYY?<;FsoO68Vs{DJDP1@;xCyerCc?^I7rvc zoDgk#5vMx7=2^5x>Hc+ClUhrJ^u43L9&OmcoaXam;WahX=<%66SGW-n#uu9M%#^mncLl+UFiSYoVlTxPX` zJ7ir^OJiird6>#SCbv2<@M>l-+Rq(*kD+pIIny?Jui^YUB9w1>DfZfnYXf|;l+OyJ zZ*S_+jKnH}-9*)I&4tmBn=6bc8CqrHDm)33*B?3P2$jR<%g>Sl3GNdgcE~4&QAE5Y 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