Write abstract and introduction around the constructive 4-coloring motivation

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>

papers/even_level_graph_generators/paper.tex

@@ -80,12 +80,88 @@
 \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)