Frame the 4-coloring motivation and add an online greedy check

Adds a Motivation section to paper.tex explaining that the
quadrilateral sequencing is intended to support an inductive 4-coloring
of the underlying maximal planar graph, with ring completion as the
suspected obstacle.

Adds commentary.tex recording (a) why a pure pigeonhole argument is
unlikely to close the conjecture, (b) the observation that under any
strictly local online rule every G'-edge constraint is enforced when
its second endpoint is colored (so ring completions cannot fail at the
moment they fire), and (c) the empirical finding that pure greedy
fails at non-ring-completion moves on every 3-connected triangulation
of order 5-7.

Adds quad_sequence_coloring_check.py, an enumeration check over small
triangulations via Sage's planar_graphs that runs greedy online
4-coloring under the canonical sequence and classifies failures.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/plane_depth_sequencing/paper.tex

@@ -36,6 +36,8 @@
 
 \documentclass{amsart}
 
+\usepackage{amssymb}
+
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}[theorem]{Lemma}
 
@@ -65,17 +67,26 @@
 
 \subjclass[2010]{Primary }
 
-\keywords{}
+\keywords{plane graph, triangulation, plane depth, deep embedding, quadrilateral decomposition, $k$-outerplanar graph, canonical sequencing}
 
 \date{}
 
 \dedicatory{}
 
 \begin{abstract}
+Given a plane embedding of a graph with outer cycle $C$, the \emph{plane depth} of a vertex is its graph distance to $C$. We use this depth function to organize plane triangulations into a layered combinatorial structure. First, we show that the subgraph induced by each depth level is outerplanar, recovering Baker's notion of a $k$-outerplanar graph. We then introduce the \emph{deep embedding} of a maximal planar graph, obtained by inserting a new vertex into each neutral triangular face (including the outer face), and prove that every face of the resulting spherical triangulation is an \emph{up} or \emph{down} triangle. Pairing adjacent triangles across their unique level edge yields a \emph{quadrilateral decomposition} into three combinatorial types: shallow diamonds, deep diamonds, and S quads. Finally, we define a deterministic traversal of this decomposition using four moves --- anchor drop, level add, join, and ring completion --- under a fixed precedence, and prove that, starting from any boundary deep diamond, the resulting sequence visits every quadrilateral exactly once.
 \end{abstract}
 
 \maketitle
 
+\section{Motivation}
+
+This paper is one step of a longer programme aimed at 4-coloring maximal planar graphs by inductive local construction. The quadrilateral sequencing is designed so that the deep embedding $G'$ can be built up one quadrilateral at a time, and so that --- at least at first glance --- each new quadrilateral admits a natural local 4-coloring choice for the vertices it introduces.
+
+Three of the four moves --- anchor drop, level add, and join --- attach a quadrilateral that introduces one or two new vertices. For each of these moves the new vertex (or vertices) can be assigned a color that respects the local triangle constraints. The fourth move, \emph{ring completion}, introduces no new vertices: it attaches a quadrilateral all four of whose corners are already in the slice. Whether the move succeeds therefore depends on previously chosen colors.
+
+A natural question is whether the local freedom present at the non-ring-completion moves is enough to guarantee that, by the time each ring completion fires, the four corner colors are already compatible with a proper 4-coloring of the new quadrilateral. The remaining sections develop the structural machinery needed to phrase this question precisely; companion commentary (\texttt{commentary.tex}) records what is known so far about why this hope is delicate, and an empirical check (\texttt{quad\_sequence\_coloring\_check.py}) explores the behavior of a particular online coloring discipline on small triangulations.
+
 \section{Definitions}
 
 \begin{definition}