On the Face Vectors of Arrangements of Curves

We consider the realizations of a sequence \((p^*_3, p^*_5, p^*_6, \ldots)\) of nonnegative integers satisfying the equation \(\sum_{k\geq 3} (k-4)p_k + 8 = 0\) 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.