didericis
435f055d82
New paper introducing the dual (inner/weak dual) of a maximal planar graph, dual depth (BFS-derived min level over a face's vertices), and a Tait-based framing of a minimal 4CT counterexample via nested level duals. Includes a dual-depth figure and its generator. Shelved per closing note in favour of an alternative approach. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
122 lines
4.2 KiB
TeX
122 lines
4.2 KiB
TeX
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\begin{document}
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\title{Nested Level Duals}
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% author one information
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\author{Eric Bauerfeld}
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\address{}
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\curraddr{}
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\email{}
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\thanks{}
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\subjclass[2010]{Primary }
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\keywords{plane graph, triangulation, plane depth, level edge, dual graph}
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\date{}
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\dedicatory{}
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\begin{abstract}
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% TODO: abstract.
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\end{abstract}
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\maketitle
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\section{Introduction}
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A classical theorem of Tait recasts the Four Colour Theorem in dual,
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edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable
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if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a
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minimal counterexample to the Four Colour Theorem -- a smallest triangulation
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admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
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admitting no proper $3$-edge-colouring.
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We study the structure such a minimal counterexample would have to exhibit
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through the lens of \emph{nested level duals}. Fixing a level source $S$ in $G$
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endows the dual $G'$ with a Breadth-First-Search--derived labelling, the dual
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depth of Definition~\ref{def:dual-depth}, and the level structure of $G$ organises
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$G'$ into a family of nested cycles carrying these labels. Our aim is to express
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the obstruction to a $3$-edge-colouring of $G'$ as conditions on this nested
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labelled-cycle structure.
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Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)
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with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
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and $G$ has $2n - 4$ triangular faces.
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\begin{definition}[Level source]
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A \emph{level source} of $G$ is any vertex $v \in V$; we write
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$S = \{v\}$ for the level-0 source.
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\end{definition}
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\begin{definition}[Levels]
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Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is
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$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest
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source vertex.
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\end{definition}
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\begin{definition}[Dual]
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The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with
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respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face
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$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by
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two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex,
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and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a
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triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$
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of $G$, and we write $V(f) \subseteq V$ for its three incident vertices.
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\end{definition}
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\begin{definition}[Dual depth]
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\label{def:dual-depth}
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Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex
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$d_f \in V(G')$ is
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\[
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\delta_G(d_f) = \min_{v \in V(f)} \ell_G(v)
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= \min_{v \in V(f)} \mathrm{dist}_G(v, S),
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\]
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the smallest level among the three vertices of $G$ bounding the face $f$.
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\end{definition}
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.7\textwidth]{fig_dual_depth.png}
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\caption{Dual depth in a stacked-ring triangulation $G$ with level source
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$S = \{0\}$. Each $G$ vertex is labelled by its level $\ell$. Each bounded face
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carries a dual vertex (square, joined by dashed dual edges) coloured by its dual
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depth $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: the central fan has depth $0$,
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the inner annulus depth $1$, and the outer annulus depth $2$. The outer face
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(the level-$3$ triangle) is excluded from the inner dual and carries no dual
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vertex.}
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\label{fig:dual-depth}
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\end{figure}
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\end{document}
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% NOTE (2026-05-22): This paper is being shelved in favour of an alternative
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% approach. The nested-level-duals framing is preserved here for reference but
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% is not being actively developed.
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