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

Abstract

The necessary and sufficient condition that the latin square formed by the Cayley multiplication table of a group has an orthogonal mate is that the group has a complete mapping. Here, we define two generalizations of the concept of a complete mapping and show how these generalizations are related to sequenceable groups and JR-sequenceable groups respectively and that together they permit a complete characterization of left neofields. In the second part of the paper, we shall show that these generalizations also yield new constructions of block designs of Mendelsohn type.Introduction. In [11] H. B. Mann introduced the concept of a complete mapping of a finite group (G, •) and showed that, when a group has such a complete mapping, the latin square formed by its Cayley multiplication table has a transversal and, hence, an orthogonal mate. See also §1.4 of [1]. Later, L. J. Paige [13] showed that complete mappings can also be used in the construction of neofields. This fact has been used extensively in [6].More recently, it has been shown in [10] and [2] that a necessary condition for a finite group to be i?-sequenceable is that it possess a complete mapping. Similarly, we prove below that a necessary condition for a finite group to be sequenceable is that it possess a near complete mapping. In general, both sequenceable and i?-sequenceable groups permit the construction of neofields and the neofields so constructed are of special type.In this first part of our paper, we define two generalizations of the concept of a complete mapping to be called a (K, λ) complete mapping and a (K, λ) near complete mapping, respectively, and we show that all of the above-mentioned concepts can be described in terms of these generalizations. We are also able to give a complete characterization for all left neofields. In the second part (to appear shortly), we show that these generalizations also yield new constructions of block designs of Mendelsohn type whose automorphism group contains a specified subgroup and we describe in more detail how (K, λ) generalized complete mappings and near complete mappings may be constructed.

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

“…Consequently, for the corresponding /-fold perfect (v, k, λ)-MD constructed as in Theorem 5.4 we must have λ(υ -1) = 0 (mod A:). The analogous result for an /-fold perfect (k,λ) near complete mapping of a group of order v -1 is λv = 0 mod k since in this case we have λ(k -1) + sk '= λ(v -1), where s is as in Definition 2.4 of [10]. So, for the corresponding /-fold perfect (v, k, λ)-MD constructed as in Theorem 5.4, we have λv = 0 mod k in this case.…”

“…We note that a 1-fold perfect (k, λ) complete mapping is a (k,λ) complete mapping as given in Definition 2.5 of [10]. DEFINITION 5.8.…”

“…(Three of these latter were used to construct our Example 9.) From Definition 2.5 of [10], it follows that, for an /-fold perfect (k, λ) complete mapping of a group (G, •) of order v, k divides λ(υ -1). Consequently, for the corresponding /-fold perfect (v, k, λ)-MD constructed as in Theorem 5.4 we must have λ(υ -1) = 0 (mod A:).…”

“…Then the cyclic sequence (a b ) must continue as follows : a, b, a~ιb, b~x(a~ιb) The design admits a regular group of automorphisms isomorphic to (8) The sequencing 0 3 2 1 of the cyclic group C 4 defines the (5,5,1)-MD whose blocks are (0 3 1 2 oo), (1 0 2 3 oo), (2 1 3 0 oo) and (3201 oo). (See Theorem 2.2 of [10].) (9) The (14,3) complete mapping of the cyclic group C 15 given below is 2-fold perfect.…”

Vertex graceful labeling of some classes of graphs

“…The book edited by Acharia, Arumugam and Rosa [1] includes a variety of labeling methods. Hsu and Keedwell [9,10] introduced and studied the extension of graceful labeling of directed graphs. The relationship between graceful directed graphs and a variety of algebraic structures including cyclic difference sets, sequenceable groups, generalized complete mappings, near-complete mapping and neofield is discussed in [3] and [4].…”