On the Face Vectors of Arrangements of Curves

Abstract

We consider the realizations of a sequence $(p_3^{\ast },p_5^{\ast },p_6^{\ast },\dots )$ of nonnegative integers satisfying the equation $\sum _{k\ge 3}(k-4)p_k+8=0$ as an arrangement of simple curves defined by $B$. Grünbaum [4]. In this paper, we show that an Eberhard-type theorem for a digon-free arrangement of simple curves is not valid in general, while some sequences are realizable as a digon-free arrangement of simple curves.