A new upper bound for angular resolution

The angular resolution of a planar straight-line drawing of a graph is the smallest angle formed by two edges incident to the same vertex. Garg and Tamassia (ESA '94) constructed a family of planar graphs with maximum degree $d$ that have angular resolution $O((\log d{)}^{\frac{1}{2}}\mathrm{/}{d}^{\frac{3}{2}})$ in any planar straight-line drawing. This upper bound has been the best known upper bound on angular resolution for a long time. In this paper, we improve this upper bound. For an arbitrarily small positive constant $\epsilon$, we construct a family of planar graphs with maximum degree $d$ that have angular resolution $O((\log d{)}^{\epsilon }\mathrm{/}{d}^{\frac{3}{2}})$ in any planar straight-line drawing.