Maps of p-gons with a ring of q-gons

Abstract

We study 3-valent maps $M_n(p,q)$ consisting of a ring of $n$ $q$-gons whose the inner and outer domains are filled by $p$-gons, for $p,q\ge 3$. We describe a domain in the space of parameters $p$, $q$, and $n$, for which such a map may exist. With four infinite sequences of maps - prisms $M_p(p\ge 3,4)$, $M_4(4,q\ge 4)$, $M_4(5,5t+2\ge 7)$, $M_4(5,5t+3\ge 8)$, we give 20 sporadic ones. The maps whose $p$-gons form two paths are first two infinite sequences and 5 maps: $M_{28}(7,5)$, $M_{12}(6,5)$, $M_{10}(5,6)$, $M_{20}(5,7)$, $M_2(3,6)$

11 pages. to appear in Bulletin of the Institute of Combinatorics and its Applications