  No proof of the 4-color conjecture reveals why it is true; the goal has not
been to go beyond proving the conjecture. The standard approach involves
constructing an unavoidable finite set of reducible configurations to
demonstrate that a minimal counterexample cannot exist. We study the 4-color
problem from a different perspective. Instead of planar triangulations, we
consider near-triangulations of the plane with a face of size 4; we call any
such graph an a-graph. We state an a-graph coloring problem equivalent to the
4-color problem and then derive a coloring condition that a minimal a-graph
counterexample must satisfy, expressing it in terms of equivalence classes
under Kempe exchanges. Through a systematic search, we discover a family of
a-graphs that satisfy the coloring condition, the fundamental member of which
has order 12 and includes the Birkhoff diamond as a subgraph. Higher-order
members include a string of Birkhoff diamonds. However, no member has an
applicable parent triangulation that is internally 6-connected, a requirement
for a minimal counterexample. Our research suggests strongly that the coloring
and connectivity conditions for a minimal counterexample are incompatible;
infinitely many a-graphs meet one condition or the other, but we find none that
meets both.
