Biembeddings of cycle systems using integer Heffter arrays

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, for n ≡ 1 (mod 4) and k ≡3(mod 4), n k ≫ ⩾ 7 and when n ≡ 0(mod 3) then k ≡ 7(mod 12), there exist face 2-colorable embeddings of the complete graph K₂ₙₖ₊₁ onto an orientable surface where each face is a cycle of a fixed length k. In these embeddings the vertices of K₂ₙₖ₊₁ will be labeled with the elements of Z₂ₙₖ₊₁ in 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 modulo 2nk + 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 arrays H (n ; k) that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, for k ⩾ 11, that there are at least (n − 2)[((k − 11)/4)!/ ]² such nonequivalent H (n ; k) that satisfy both conditions (1) and (2)

On The Spectrum of Minimal Defining Sets of Full Designs

A defining set of a t-(v, k, λ) design is a subcollection of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subcollections is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v, k, λ) design D = (V,B) is called a full design if B is the collection of all possible k-subsets of V. Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a defining set of the corresponding t-design. With this motivation, in this paper, we study the full designs when t = 2 and k = 3 and we give several families of non-isomorphic minimal defining sets of full designs. Also, it is proven that there exist values in the spectrum of the full design on v elements such that the number of non-isomorphic minimal defining sets on each of these sizes goes to infinity as v→ ∞. Moreover, the lower bound on the size of the defining sets of the full designs is improved by finding the size of the smallest defining sets of the full designs on eight and nine points. Also, all smallest defining sets of the full designs on eight and nine points are classified

On defining sets of full designs with block size three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. 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}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d D of D there exists a minimal defining set d F of F such that $${d_D = d_F\cap D}$$. The unique simple design with parameters $${{\left(v,k, {v-2\choose k-2}\right)}}$$ is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously

Defining Sets of Full Designs with Block Size Three II

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