A list is given of all semisymmetric (edge-but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3, 3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K 1,3 as a normal quotient is investigated in detail.

The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

Let C G (X ) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G ≤ Aut X of automorphisms lifts. There is a natural lattice structure on C G (X ), where ℘ 1 ≤ ℘ 2 whenever ℘ 2 factors through ℘ 1 . The sublattice C G (℘) of coverings which are below a given covering ℘ :X → X naturally corresponds to a lattice N G (℘) of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let C p G (X ) ≤ C G (X ) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs C p G (X ) in the sense that, for each member of C p G (X ), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.

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A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs:(i) cycles C 2n , n 3; (ii) complete graphs K 2n , n 3; (iii) complete bipartite graphs K n,n , n 3; (iv) complete bipartite graphs minus a matching K n,n − nK 2 , n 3; (v) incidence and nonincidence graphs B(H 11 ) and B (H 11 ) of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs B (PG(d, q)) and B (PG(d, q)), with d 2 and q a prime power, of projective spaces; (vii) and an infinite family of regular Z d -covers K 2d q+1 of K q+1,q+1 − (q + 1)K 2 , where q 3 is an odd prime power and d is a divisor of q−1 2 and q − 1, respectively, depending on whether q ≡ 1 (mod 4) or q ≡ 3 (mod 4), obtained by identifying the vertex set of the base graph with two copies of the projective line PG (1, q), where the missing matching consists of all pairs of the form [i, i ], i ∈ PG(1, q), and the edge [i, j ] carries trivial voltage if i = ∞ or j = ∞, and carries voltageh ∈ Z d , the residue class of h ∈ Z, if and only if i − j = θ h , where θ generates the multiplicative group F * q of the Galois field F q .

It is proved that for each compact (bordered) surface 7 and each integer k there is a constant N with the following property: If 1 is a family of pairwise nonhomotopic closed curves on 7 such that any two curves from 1 intersect in at most k points, then 1 contains at most N curves.

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It is shown that a vertex-transitive graph of valency p + 1, p a prime, admitting a transitive action of a {2, p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605-615]).Let G be a permutation group on a finite set V . A non-identity element of G is semiregular if it has all orbits of equal size. It is known that each finite transitive permutation group contains a fixed-point-free element of prime power order [4, Theorem 1], but not necessarily a fixedpoint-free element of prime order, that is, a semiregular element of prime order. A permutation E-mail address: dragan.marusic@guest.arnes.si (D. Marušič).