For all integers m, nand t, we determine necessary and sufficient conditions for the existence of (1) a pair of 3-cube decompositions of Kn having precisely t common 3-cubes; and (2) a pair of 3-cube decompositions of Km,n having precisely t common 3-cubes.

A unit cube in k dimensional space (or k-cube in short) is defined as the Cartesian product R1 × R2 × · · · × Rk where Ri(for 1 ≤ i ≤ k) is a closed interval of the form [ai, ai + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection....

Hardy was surprised by Ramanujan’s remark about a London taxi numbered 1729: “it is a very interesting number, it is the smallest number expressible as a sum of two cubes in two different ways”. In memory of this story, this number is now called Taxicab(2) = 1729 = 9 + 10 = 1 + 12, Taxicab(n) being the smallest number expressible in n ways as a sum of two cubes. We can generalize the problem by...

Graphs are used in modeling interconnections networks and measuring their properties. Knowing and understanding the graph theoretical/combinatorial properties of the underlying networks are necessary in developing more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms [1] The hypercube is one of the most versatile and efficient networks yet discovered for ...

Due to its topological generality and flexibility, the k-ary n-cube architecture has been actively researched for various applications. However, the processor allocation problem has not been adequately addressed for the k-ary n-cube architecture, even though it has been studied extensively for hypercubes and meshes. The earlier k-ary n-cube allocation schemes based on conventional slice partiti...

Hypercubes are famous interconnection networks. The shuffle cubes are proposed as its variation with lower diameter. In this paper, we concentrate on the fault tolerant hamiltonicity and hamiltonian connectivity of the shuffle cube. We say that a graph G is f-hamiltonian if there is a hamiltonian cycle in G − F and is f-hamiltonian connected if there is a hamiltonian path between any two vertic...

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T , we prove that the universal invariant JT of T associated to the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2 is contained in...

We study the problem of acute triangulations of convex polyhedra and the space Rn. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n–cube do not exist for n ≥ 4. Further, we prove that acute triangulations of the space Rn do not exist for n ≥ 5. In the opposite direction, in R3, we present a construction of...

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T , we prove that the universal invariant JT of T associated to the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2 is contained in...

Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N = 24, 26, ≥ 28; 7-designs for N = 24, 30, 32, 34, ≥ 36; 8-designs for N = 36, 40, 42, ≥ 44; 9-designs for N = 48, 50, 52, ≥ 54; 10-designs for N = 60, 62, ≥ 64; 11-designs for N = 70, 72, ≥ 74; and 12-designs for N = 84, ≥ 86. The existence of some of these designs is established analytical...