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 $k \geq 5$. Also vertex-minimal non-singular $\{ p, p\}$-pattern are constructed for all odd primes $p$.
Categories: math.GT math.CO