Regular maps from Cayley graphs III: tt-balanced Cayley maps

The class of t-balanced Cayley maps [J. Martino, M. Schultz, Symmetrical Cayley maps with solvable automorphism groups, abstract in SIGMAC ’98, Flagstaff, AR, 1998] is a natural generalisation of balanced and antibalanced Cayley maps introduced and studied by Širáň and Škoviera [Regular maps from Cayley graphs II: antibalanced Cayley maps, Discrete Math. 124 (1994) 179–191; Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189–202]. The present paper continues this study by investigating the distribution of inverses, automorphism groups, and exponents of t-balanced Cayley maps. The methods are based on the use of t  -automorphisms of groups with sign structure which extend the notion of an antiautomorphism crucial for antibalanced Cayley maps. As an application, a new series of nonstandard regular embeddings of complete bipartite graphs Kn,nKn,n is constructed for each n divisible by 8.