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

Original · EN

A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In 1983, 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 2001 that there exist infinitely many {4, 4}- and {3, 6}-equivelar polyhedral maps. In 1990, 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}-pattern are constructed for all odd primes p.