Partitions of groups and complete mappings

Generalized complete mappings, neofields, sequenceable groups and block designs. II

“…On the other hand, it is proved in [5] that a cyclic (v, 3,1)-MD exists if and only if v = 1 or 3 (mod 6), v Φ 9. It is conjectured (see [7] and [8]) that (3,1) complete mappings of the cyclic group C υ exist for all v = 1 (mod 6). If this were proved, we should then have that cyclic (v, 3, l)-RMD's exist if and only if v = 1 (mod 6) (the other direction of the implication is easy to see).…”

Coset-wise affine functions and cycle types of complete mappings

“…(1) A complete mapping (or orthormorphism) of cycle type x 1 x (q−1)/ℓ ℓ is called ℓregular (see [9], whence this terminology apparently originated). Because a scalar multiplication F q → F q , x → ax for fixed a ∈ F q , is a complete mapping of F q if and only if a / ∈ {0, −1}, we see that ℓ-regular complete mappings of F q exist for every divisor ℓ of q − 1 such that…”

Graceful labeling of digraphs—a survey

“…If a degree balanced digraph D is graceful then number of arcs in D is even.By using Friedander et al[8] results Hegde and Shivarajkumar[12] proved the following Theorem 4.16. The digraph Dðn 1 , n 2 , :::, n k ; m 1 , m 2 , :::, m k Þ is graceful if and only if P k i¼1 n i m i is even.…”