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In this paper, we consider the uniformly resolvable decom-positions of the complete graph K-v minus a 1-factor (K-v - I) into sub -graphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars., The first and the third authors are supported by I.N.D.A.M (G. N. S. A. G. A.) , Italy.

Square Heffter arrays are n×n arrays such that each row and each column contains k filled cells, each row and column sum is divisible by 2nk+1 and either x or −x appears in the array for each integer 1⩽x⩽nk. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k≡0(mod4); or (b) n≡1(mod4) and k≡3(mod4); or (c) n≡0(mod4), k≡3(mod4) and n≫k. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.

In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.

In this paper, we use constructions of Heffter arrays to verify the existence of face 2-colorable embeddings of cycle decompositions of the complete graph. Specifically, forn degrees 1(mod 4)andk degrees 3(mod4),n >> k > 7and whenn degrees 0(mod 3)thenk degrees 7(mod 12), there exist face 2-colorable embeddings of the complete graphK2nk+1onto an orientable surface where each face is a cycle of a fixed lengthk. In these embeddings the vertices ofK2nk+1will be labeled with the elements ofZ2nk+1in such a way that the group,(Z2nk+1,+)acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays,H(n;k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo2nk+1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arraysH(n;k)that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, fork > 11, that there are at least(n-2)[((k-11)/4)!/e]2such nonequivalentH(n;k)that satisfy both conditions (1) and (2).

In this paper, we use constructions of Heffter arrays to verify the existence of face 2-colorable embeddings of cycle decompositions of the complete graph. Specifically, forn degrees 1(mod 4)andk degrees 3(mod4),n >> k > 7and whenn degrees 0(mod 3)thenk degrees 7(mod 12), there exist face 2-colorable embeddings of the complete graphK2nk+1onto an orientable surface where each face is a cycle of a fixed lengthk. In these embeddings the vertices ofK2nk+1will be labeled with the elements ofZ2nk+1in such a way that the group,(Z2nk+1,+)acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays,H(n;k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo2nk+1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arraysH(n;k)that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, fork > 11, that there are at least(n-2)[((k-11)/4)!/e]2such nonequivalentH(n;k)that satisfy both conditions (1) and (2)., Scientific and Technological Research Council of Turkey (TÜBİTAK) 2219; University of Queensland School of Mathematics and Physics, Ethel Raybould Visiting Fellowship

A Heffter array H(n; k) is an n x n matrix such that each row and column contains k filled cells, each row and column sum is divisible by 2nk + 1 and either x or -x appears in the array for each integer 1 <= x <= nk. Heffter arrays are useful for embedding the graph K-2(nk+1) on an orientable surface. An integer Heffter array is one in which each row and column sum is 0. Necessary and sufficient conditions (on n and k) for the existence of an integer Heffter array H(n; k) were verified by Archdeacon, Dinitz, Donovan and Yazici (2015) and Dinitz and Wanless (2017). In this paper we consider square Heffter arrays that are not necessarily integer. We show that such Heffter arrays exist whenever 3 <= k < n., NA

For an m-cycle C, an inside m-cycle of C is a cycle that is on the same vertex set and edge-disjoint from C. In an m-cycle system, (X, C), if inside m-cycles can be chosen-one for each cycle-to form another m-cycle system, then (chi, C) is called an inside perfect m-cycle system. Inside perfect cycle systems can be considered as generalisations of i-perfect cycle systems. Cycle packings are generalisations of cycle systems that may have leaves after decomposition. In this paper, we prove that an inside perfect maximum packing of K-n with 8-cycles of order n exists for each n >= 8. We also construct a maximum 8-cycle packing of order n which is not inside perfect for each n >= 10., NA

In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal Array-based Latin Hypercube sampling is the same. We then extend these results to an experimental design setting by projecting onto a t < d dimensional subspace. These results are confirmed by simulations. The theory presented has both theoretical and practical significance in modelling and simulation science when sampling over high dimensional spaces., University of Queensland Centre for Coal Seam Gas; APLNG; Arrow Energy; Santos; QGC (Shell); Scientific and Technological Research Council of Turkey (TÜBİTAK)

We show that a pair of orthogonal partial Latin squares of order n can be embedded in a pair of orthogonal Latin squares of order at most 16n(4) and all orders greater than or equal to 48n(4). This paper provides the first direct polynomial order embedding construction for pairs of orthogonal partial Latin squares., NA

A flower, F-s(x), around a point x in a Steiner triple system D = (V, B) is the set of all triples in B which contain the point x, namely F-D(X)={b is an element of B vertical bar x is an element of b}. This paper determines the possible number of common flowers that two Steiner triple systems can have in common. For all admissible pairs (k, v) where k <= v-6 we construct a pair of Steiner triple systems of order v where the flowers around k elements of V are identical in both Steiner triple systems, except for the pairs (2, 9), (3, 9) and (6, 13). Equivalently this result shows that there is a Steiner triple trade of foundation I = v k that can be embedded in a STS(v) for each admissible v and 6 <= l <= v except when (l, v) = (6, 9), (7, 9) or (7, 13)., Scientific and Technological Research Council of Turkey (TÜBİTAK)

A defining set of a t-(v, k, lambda) design is a subcollection of its blocks which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. The unique simple design with parameters is said to be the full design on v elements. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. The largest known minimal defining set is given. The existence of a continuous section of the spectrum comprising asymptotically 9v (2)/50 values is shown. This gives a quadratic length section of continuous spectrum where only a linear section with respect to v was known before., Raybould fellowship; Scientific and Technological Research Council of Turkey (TÜBİTAK) Career

We prove that any complete multipartite graph with parts of even size can be decomposed into closed trails with prescribed even lengths., NA