diff --git a/papers/nested_level_duals/experiments/draw_dual_depth.py b/papers/nested_level_duals/experiments/draw_dual_depth.py new file mode 100644 index 0000000..7f5d1d2 --- /dev/null +++ b/papers/nested_level_duals/experiments/draw_dual_depth.py @@ -0,0 +1,154 @@ +"""Draw a diagram illustrating dual depth. + +We build a concentric ("stacked rings") triangulation G with a single level +source S = {0} at the centre, so the vertices fall into clean BFS levels +0, 1, 2, 3 by radius. We then overlay the inner (weak) dual G' -- one dual +vertex per bounded triangular face -- and colour each dual vertex by its dual +depth: the minimum level among the three vertices of the corresponding face. + +The construction makes all three attainable dual depths (0, 1, 2) appear; the +outer face (the level-3 triangle) is excluded from the inner dual. +""" +import math +import os +import networkx as nx +import matplotlib.pyplot as plt +from matplotlib.lines import Line2D + +OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__))) + +# --------------------------------------------------------------------------- +# Construct G: centre 0 (level 0) plus three triangular rings at radii 1,2,3. +# Vertex j of every ring sits at angle 90 + 120*j degrees, so spokes are radial. +# --------------------------------------------------------------------------- +RINGS = 3 # ring 3 is the outer-face triangle +pos = {0: (0.0, 0.0)} +ring = {0: [0]} # ring index -> vertex ids +nxt = 1 +for r in range(1, RINGS + 1): + ids = [] + for j in range(3): + ang = math.radians(90 + 120 * j) + pos[nxt] = (r * math.cos(ang), r * math.sin(ang)) + ids.append(nxt) + nxt += 1 + ring[r] = ids + +G = nx.Graph() +G.add_nodes_from(pos) + +# centre fan +for v in ring[1]: + G.add_edge(0, v) +# ring triangles +for r in range(1, RINGS + 1): + a, b, c = ring[r] + G.add_edges_from([(a, b), (b, c), (c, a)]) +# radial spokes + annulus diagonals (inner_j -- outer_{j+1}) +for r in range(1, RINGS): + inner, outer = ring[r], ring[r + 1] + for j in range(3): + G.add_edge(inner[j], outer[j]) # spoke + G.add_edge(inner[j], outer[(j + 1) % 3]) # diagonal + +assert G.number_of_edges() == 3 * G.number_of_nodes() - 6, "not a triangulation" + +# --------------------------------------------------------------------------- +# Levels: BFS distance from the source S = {0}. +# --------------------------------------------------------------------------- +S = {0} +level = nx.multi_source_dijkstra_path_length(G, S) + +# --------------------------------------------------------------------------- +# Bounded faces (the inner dual's vertices). Listed explicitly from the +# construction; the outer face (ring 3 triangle) is omitted. +# --------------------------------------------------------------------------- +faces = [] +a, b, c = ring[1] +faces += [(0, a, b), (0, b, c), (0, c, a)] # centre fan +for r in range(1, RINGS): + inn, out = ring[r], ring[r + 1] + for j in range(3): + i0, i1 = inn[j], inn[(j + 1) % 3] + o1 = out[(j + 1) % 3] + o0 = out[j] + faces += [(i0, i1, o1), (i0, o1, o0)] # the two annulus triangles + +def dual_depth(face): + return min(level[v] for v in face) + +# dual vertex positions = face centroids +dpos = {} +ddepth = {} +for f in faces: + cx = sum(pos[v][0] for v in f) / 3.0 + cy = sum(pos[v][1] for v in f) / 3.0 + dpos[f] = (cx, cy) + ddepth[f] = dual_depth(f) + +# dual edges: two bounded faces sharing an edge of G +edge_faces = {} +for f in faces: + for i in range(3): + e = frozenset((f[i], f[(i + 1) % 3])) + edge_faces.setdefault(e, []).append(f) +dual_edges = [fs for fs in edge_faces.values() if len(fs) == 2] + +# --------------------------------------------------------------------------- +# Draw. +# --------------------------------------------------------------------------- +LEVEL_COLOR = {0: '#1e293b', 1: '#475569', 2: '#94a3b8', 3: '#cbd5e1'} +DEPTH_COLOR = {0: '#16a34a', 1: '#2563eb', 2: '#dc2626'} + +fig, ax = plt.subplots(figsize=(9, 9)) + +# G edges (light) and vertices (labelled by level) +nx.draw_networkx_edges(G, pos, ax=ax, edge_color='#d1d5db', width=1.4) +for v, (x, y) in pos.items(): + ax.scatter([x], [y], s=520, color=LEVEL_COLOR[level[v]], + edgecolors='black', linewidths=1.0, zorder=3) + ax.text(x, y, f'{v}\n$\\ell{{=}}{level[v]}$', ha='center', va='center', + color='white', fontsize=8.5, fontweight='bold', zorder=4) + +# dual edges (dashed) and dual vertices (squares, coloured by dual depth) +for f0, f1 in dual_edges: + (x0, y0), (x1, y1) = dpos[f0], dpos[f1] + ax.plot([x0, x1], [y0, y1], color='#fb923c', lw=1.3, ls='--', zorder=2) +for f, (x, y) in dpos.items(): + ax.scatter([x], [y], s=300, marker='s', color=DEPTH_COLOR[ddepth[f]], + edgecolors='black', linewidths=0.8, zorder=5) + ax.text(x, y, str(ddepth[f]), ha='center', va='center', + color='white', fontsize=9, fontweight='bold', zorder=6) + +legend = [ + Line2D([0], [0], marker='o', color='w', label='$G$ vertex (label = level $\\ell$)', + markerfacecolor='#475569', markeredgecolor='black', markersize=12), + Line2D([0], [0], color='#fb923c', ls='--', lw=1.3, label="dual edge of $G'$"), + Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 0$', + markerfacecolor=DEPTH_COLOR[0], markeredgecolor='black', markersize=11), + Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 1$', + markerfacecolor=DEPTH_COLOR[1], markeredgecolor='black', markersize=11), + Line2D([0], [0], marker='s', color='w', label='dual depth $\\delta = 2$', + markerfacecolor=DEPTH_COLOR[2], markeredgecolor='black', markersize=11), +] +ax.legend(handles=legend, loc='upper left', fontsize=10, framealpha=0.95) + +ax.set_aspect('equal') +ax.axis('off') +ax.set_title("Dual depth in a stacked-ring triangulation $G$ with source $S=\\{0\\}$.\n" + "Each bounded face carries a dual vertex (square) coloured by its dual " + "depth\n$\\delta(d_f)=\\min_{v\\in V(f)}\\ell(v)$. " + "The outer face (level-3 triangle) has no dual vertex.", + fontsize=11) +fig.tight_layout() + +out = os.path.join(OUT_DIR, 'fig_dual_depth.png') +fig.savefig(out, dpi=180, bbox_inches='tight') +plt.close(fig) + +# console summary +from collections import Counter +print(f'n={G.number_of_nodes()} edges={G.number_of_edges()} ' + f'bounded faces={len(faces)} dual edges={len(dual_edges)}') +print('dual depth distribution:', dict(sorted(Counter(ddepth.values()).items()))) +print(f'wrote {out}') diff --git a/papers/nested_level_duals/fig_dual_depth.png b/papers/nested_level_duals/fig_dual_depth.png new file mode 100644 index 0000000..fc1144b Binary files /dev/null and b/papers/nested_level_duals/fig_dual_depth.png differ diff --git a/papers/nested_level_duals/paper.aux b/papers/nested_level_duals/paper.aux new file mode 100644 index 0000000..9d8e8b2 --- /dev/null +++ b/papers/nested_level_duals/paper.aux @@ -0,0 +1,11 @@ +\relax +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{0pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\@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}} +\gdef \@abspage@last{2} diff --git a/papers/nested_level_duals/paper.fdb_latexmk b/papers/nested_level_duals/paper.fdb_latexmk new file mode 100644 index 0000000..2508ddb --- /dev/null +++ b/papers/nested_level_duals/paper.fdb_latexmk @@ -0,0 +1,64 @@ +# Fdb version 3 +["pdflatex"] 1779482777 "paper.tex" "paper.pdf" "paper" 1779482778 + "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 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with amsart +%% ==================================================================== + +\documentclass{amsart} + +\usepackage{amssymb} +\usepackage{graphicx} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Nested Level Duals} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{plane graph, triangulation, plane depth, level edge, dual graph} + +\date{} + +\dedicatory{} + +\begin{abstract} +% TODO: abstract. +\end{abstract} + +\maketitle + +\section{Introduction} + +A classical theorem of Tait recasts the Four Colour Theorem in dual, +edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable +if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a +minimal counterexample to the Four Colour Theorem -- a smallest triangulation +admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph +admitting no proper $3$-edge-colouring. + +We study the structure such a minimal counterexample would have to exhibit +through the lens of \emph{nested level duals}. Fixing a level source $S$ in $G$ +endows the dual $G'$ with a Breadth-First-Search--derived labelling, the dual +depth of Definition~\ref{def:dual-depth}, and the level structure of $G$ organises +$G'$ into a family of nested cycles carrying these labels. Our aim is to express +the obstruction to a $3$-edge-colouring of $G'$ as conditions on this nested +labelled-cycle structure. + +Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) +with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ +and $G$ has $2n - 4$ triangular faces. + +\begin{definition}[Level source] +A \emph{level source} of $G$ is any vertex $v \in V$; we write +$S = \{v\}$ for the level-0 source. +\end{definition} + +\begin{definition}[Levels] +Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is +$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest +source vertex. +\end{definition} + +\begin{definition}[Dual] +The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with +respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face +$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by +two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex, +and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a +triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$ +of $G$, and we write $V(f) \subseteq V$ for its three incident vertices. +\end{definition} + +\begin{definition}[Dual depth] +\label{def:dual-depth} +Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex +$d_f \in V(G')$ is +\[ + \delta_G(d_f) = \min_{v \in V(f)} \ell_G(v) + = \min_{v \in V(f)} \mathrm{dist}_G(v, S), +\] +the smallest level among the three vertices of $G$ bounding the face $f$. +\end{definition} + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\textwidth]{fig_dual_depth.png} +\caption{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 $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: 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.} +\label{fig:dual-depth} +\end{figure} + +\end{document} + +% NOTE (2026-05-22): This paper is being shelved in favour of an alternative +% approach. The nested-level-duals framing is preserved here for reference but +% is not being actively developed.