On the complexity of arrangements of circles in the plane

Continuing and extending the analysis in a previous paper [15], we establis h several combinatorial results on the complexity of arrangements of circle s in the plane. The main results are a collection of partial solutions to t he conjecture that (a) any arrangement of unit circles with at least one in tersecting pair has a vertex incident to at most three circles, and (b) any arrangement of circles of arbitrary radii with at least one intersecting p air has a vertex incident to at most three circles, under appropriate assum ptions on the number of intersecting pairs of circles (see below for a more precise statement).