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\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.