t-Balanced Cayley maps of dihedral groups

A Cayley map is an embedding of a Cayley graph where the cyclic ordering of generators around every vertex is the same. The involution indicating the position of mutually inverse generators in the cyclic ordering is called the distribution of inverses of a Cayley map. The Cayley maps whose distribution of inverses is linear (modulo the degree of the map) with ʹslopeʹ t are called t-balanced. An exponent of a Cayley map is a number e with the property that, roughly speaking, the Cayley map is isomorphic to its ʹe-fold rotational imageʹ. contribution we present results related to the construction of t-balanced Cayley maps which are not regular and do not have t as an exponent.