A k-extended Skolem sequence of order n is an integer sequence (s,, s2,. . . , S Z~+~) in which sk = 0 and for eachj E (1,. . . ,n}, there exists a unique i E (1,. . . ,2n} such that si = s i + j = j .We show that such a sequence exists if and only if either 1) k is odd and n = 0 or 1 (mod 4) or (2) k is even and n = 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. 0 1995 John Wiley & Sons, Inc.

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We give new examples of graphs with the n-e.c. adjacency property. Few explicit families of n-e.c. graphs are known, despite the fact that almost all finite graphs are n-e.c. Our examples are collinearity graphs of certain partial planes derived from affine planes of even order. We use probabilistic and geometric techniques to construct new examples of n-e.c. graphs from partial planes for all n, and we use geometric techniques to give infinitely many new explicit examples if n = 3. We give a new construction, using switching, of an exponential number of non-isomorphic n-e.c. graphs for certain orders.

A Skolem sequence of order n is a sequence S = (sl, s 2 . . . , szn) of 2n integers satisfying the following conditions: (1) for every k E {1, 2 , . . . ,n} there exist exactly two elements si,sj suchIn this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. 0 1993 John Wiley & Sons, Inc. SEQUENCES AND CYCLIC STSA Steiner triple system of order v, STS(v), is a pair of sets ( V , B ) , where IVl = v and B consists of 3-subsets (triples or blocks) of V such that any two elements of V occur in exactly one triple. An STS(v) exists if and only if v = 1 or 3 (mod 6). Two Steiner triple systems on the same set V are disjoint if they have no blocks in common. In [8,9,10,20], it is shown that the maximum number of painvise disjoint Steiner triple systems on a v-element set is v -2 for all v > 7. The maximum is 2 for v = 7.A STS(v) is cyclic if its automorphism group contains a v-cyclic. A cyclic STS (v) exists for all v = 1 or 3 (mod 6) except for v = 9 [13]. This question of existence is equivalent to finding solutions to Heffter's difference problems 161: * Research partially supported by grants from NSERC Canada.

Abstract. Although the coordinate ternary field of a topological affine plane is topological, the converse does not hold. However, an affine plane is topological precisely when its coordinate biternary fields are topological. We extend this result to topological biternary rings and their topological affine Klingenberg planes. Then we examine the locally compact situation. Finally, following the ideas of Knarr and Weigand, we show that in certain circumstances, the continuity of the ternary operators is sufficient to ensure that the biternary ring is topological. This facilitates the construction of locally compact, locally connected affine Klingenberg planes.Mathematics Subject Classifications (1991): 51E 15, 54H 13.