Resolvable minimum coverings with quadruples

Let V be a finite set of v elements. A covering of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at least one member of F. For fixed v and k, the covering problem is to determine the number of blocks in any minimum covering. A minimum covering is resolvable if we can partition the blocks into classes (called resolution classes) such that every element is contained in precisely one block of each class. A resolvable minimum covering of the pairs of V by k-subsets is denoted by RC(v, k). In this article, we show that there exist RC(v, 4) for v ≡ 0 (mod 4) except for v = 12 and possibly for v ∈ {104, 108, 116, 132, 156, 164, 204, 212, 228, 276}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 431–450, 1998     

The Doyen—Wilson theorem for minimum coverings with triples

In this article necessary and sufficient conditions are found for a minimum covering of K_m with triples to be embedded in a minimum covering of K_n with triples. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:341–352, 1997     

Branched coverings

Basic results on combinatorial branched coverings between relative geometric cycles are given. It is shown that every geometricn-cycle is a branched covering overS. If the downstairs space of a branched covering is locally simply connected then the branched set is a pure subcomplex of codimension 2. Finally, several Hurwitz-like theorems on existence and representation of branched coverings between relative geometric cycles are derived.This work was supported in part by the Research Council of Slovenia, Yugoslavia.     

Equitable resolvable coverings

In an earlier article, Willem H. Haemers has determined the minimum number of parallel classes in a resolvable 2‐(qk,k,1) covering for all k ≥ 2 and q = 2 or 3. Here, we complete the case q = 4, by construction of the desired coverings using the method of simulated annealing. Secondly, we look at equitable resolvable 2‐(qk,k,1) coverings. These are resolvable coverings which have the additional property that every pair of points is covered at most twice. We show that these coverings satisfy k < 2q ? , and we give several examples. In one of these examples, k > q. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 113–123, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10024     

Typical Frobenius coverings

A covering p from a Cayley graph Cay（G, X） onto another Cay（H, Y） is called typical Frobenius if G is a Frobenius group with H as a Frobenius complement and the map p ： G →H is a group epimorphism. In this paper, we emphasize on the typical Frobenius coverings of Cay（H, Y）. We show that any typical Frobenius covering Cay（G, X） of Cay（H, Y） can be derived from an epimorphism /from G to H which is determined by an automorphism f of H. If Cay（G, X1） and Cay（G, X2） are two isomorphic typical Frobenius coverings under a graph isomorphism Ф, some properties satisfied by Фare given.     

Large sets of coverings

Large sets of packings were investigated extensively. Much less is known about the dual problem, i.e., large sets of coverings. We examine two types of important questions in this context; what is the maximum number of disjoint optimal coverings? and what is the minimum number of optimal coverings for which the union covers the space? We give various constructions which give the optimal solutions and some good upper and lower bounds on both questions, respectively. © 1994 John Wiley & Sons, Inc.     

Automorphisms of graphs and coverings

We apply the theory of covering spaces to show how one can construct infinitely many finite s-transitive or locally s-transitive graphs. N. Biggs has used for similar purpose a special graph covering construction due to J. H. Conway.     

Path coverings with paths

In this paper, we give two new proofs of a result of Heinrich, Langdeau and Verrall that provide necessary and sufficient conditions for the existence of a set S of 3‐paths in K_n having the property that each 2‐path in K_n lies in exactly one path in S. These are then used to consider the case n ≡ 3 (mod 4) when no such exact covering is possible, and to solve the problem of covering (k−1)‐paths with k‐paths for all k ≥ 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 156–167, 2001     

Essential coverings and triangulations

Sunto In questo lavoro si danno condizioni sufficienti per un ricoprimento di una varietà differenziabile V affinchè il suo nerbo abbia lo stesso tipo P.L. di V e si prova l'esistenza di una larga classe di siffatti ricoprimenti.

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The author is member of G.N.S.A.G.A. of the C.N.R. and partially supported by a N.A.T.O.-C.N.R. fellowship.