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Write abstract and introduction around the constructive 4-coloring motivation

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Frame the paper's purpose: ask whether two constructive families of
4-colorable triangulations -- bridge-derived level graphs (parity
2-coloring) and intertwining trees (two trees, disjoint color pairs) --
suffice to generate every maximal planar graph on n vertices. An
affirmative answer would be a constructive proof of the four color theorem
for triangulations. State the duality bridge to Tait/Holton-McKay and the
n=21 confirmation.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-22 11:34:07 -04:00
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commit 5675207ae0
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papers/even_level_graph_generators/paper.aux
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\dedicatory{}
\begin{abstract}
We investigate whether two constructive families of $4$-colorable
triangulations are, for each $n$, already rich enough to produce every
maximal planar graph on $n$ vertices. The first family is the
\emph{bridge-derived level graphs}: starting from an \emph{Even Level
Graph} -- a triangulation all of whose level cycles, measured by
breadth-first distance from a chosen source vertex, are even -- we apply
\emph{bridge switches}, edge switches that never close a cycle in either
parity subgraph. Every such graph is $4$-colorable, inheriting the parity
$2$-coloring of an Even Level Graph. The second family is the
\emph{intertwining trees}: triangulations whose vertices split into two
sets each inducing a tree, which are $4$-colorable by coloring the two
trees from disjoint pairs of colors. We conjecture that every maximal
planar graph is a bridge-derived level graph, an intertwining tree, or
both; since both families are $4$-colorable by construction, the
conjecture would give a constructive proof of the four color theorem for
triangulations, and hence for all planar graphs. We show that a
triangulation is an intertwining tree exactly when its dual is
Hamiltonian, so every triangulation on at most $20$ vertices is an
intertwining tree and the first possible failures occur at $n = 21$, at
the six duals of the Holton--McKay graphs. We verify that all six are
bridge-derived level graphs, confirming the conjecture in its first
nontrivial case.
\end{abstract}
\maketitle
\section{Introduction}
The four color theorem states that every planar graph is properly
$4$-colorable. It suffices to prove this for \emph{maximal} planar graphs
(triangulations), since every planar graph is a spanning subgraph of one.
The known proofs proceed by reducing a hypothetical minimal counterexample,
either through computer-checked unavoidable configurations or through
discharging.
We take a constructive view. Instead of coloring an arbitrary
triangulation, we ask which triangulations can be \emph{assembled} by
operations that manifestly preserve $4$-colorability, and whether those
operations reach all of them. We study two such constructions, and our
motivating question is whether the two together are sufficient: does every
maximal planar graph on $n$ vertices arise from one of them?
The first construction builds on the \emph{level} structure of a
triangulation. Fixing a source vertex and taking breadth-first levels, an
\emph{Even Level Graph} (Definition~\ref{def:even-level-graph}) is a
triangulation whose level cycles are all even; equivalently both of its
parity subgraphs are bipartite, and a $2$-coloring of each parity subgraph
-- two colors for the even-level vertices, two for the odd -- is a proper
$4$-coloring (Theorem~\ref{thm:even-level-4colorable}). From an Even Level
Graph we generate further triangulations by edge switches; restricting to
\emph{bridge switches} (Definition~\ref{def:bridge-switch}), which add an
edge to a parity subgraph only when it is a bridge there, guarantees that
no new cycle -- and in particular no odd cycle -- ever appears in a parity
subgraph. The resulting \emph{bridge-derived level graphs}
(Definition~\ref{def:bridge-derived-level-graph}) therefore remain
$4$-colorable by the same parity coloring.
The second construction is purely combinatorial: an \emph{intertwining
tree} (Definition~\ref{def:intertwining-tree}) is a triangulation whose
vertex set partitions into two parts each inducing a tree. Coloring one
tree from $\{1,2\}$ and the other from $\{3,4\}$ is a proper $4$-coloring,
since edges inside a part join differently-colored tree vertices and edges
across the parts join the disjoint color sets.
Our central question is whether these two families exhaust all
triangulations
(Conjecture~\ref{conj:every-triangulation-derived}). As both families
consist of $4$-colorable graphs, an affirmative answer would constitute a
constructive proof of the four color theorem for triangulations.
We connect the two constructions through duality: a triangulation is an
intertwining tree if and only if its dual is Hamiltonian
(Theorem~\ref{thm:intertwining-iff-hamiltonian-dual}). Tait's conjecture --
that every $3$-connected cubic planar graph is Hamiltonian -- fails first
at $38$ vertices, where Holton and McKay found exactly six counterexamples;
dually, every triangulation on at most $20$ vertices is an intertwining
tree, and the first triangulations that are not are the six $21$-vertex
duals of the Holton--McKay graphs. These six are therefore the first
nontrivial test of the conjecture, and we verify that all six are
bridge-derived level graphs -- each at most four bridge switches from an
Even Level Graph, and two of them Even Level Graphs already.
\section{Definitions}
Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation)