  In the first partial result toward Steinberg's now-disproved three coloring
conjecture, Abbott and Zhou used a counting argument to show that every planar
graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their
proof is a fact about plane graphs: in any plane graph of minimum degree 3, if
no two triangles share an edge, then triangles make up strictly less than 2/3
of the faces. We show how this result, combined with Kostochka and Yancey's
resolution of Ore's conjecture for k = 4, implies that every planar graph
without cycles of lengths 4 through 8 is 3-colorable.
