A note on the existence of {k, k}-equivelar polyhedral maps

Abstract

A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p >q = 4 or q - 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}- and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}- and {6, 6}-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual {k, k}-equivelar polyhedral maps exist for each k ≥ 5. Also vertex-minimal non-singular {p, p}-patterns are constructed for all odd primes p.