Characterizing which powers of hypercubes and folded hypercubes are divisor graphs

Characterizing which powers of hypercubes and folded hypercubes are divisor graphs

In this paper, we show that Q n is a divisor graph, for n = 2, 3. For n ≥ 4, we show that Q n is a divisor graph iff k ≥ n− 1. For folded-hypercube, we get FQn is a divisor graph when n is odd. But, if n ≥ 4 is even integer, then FQn is not a divisor graph. For n ≥ 5, we show that (FQn) k is not a divisor graph, where 2 ≤ k ≤ ⌈ 2 ⌉ − 1.

Cycles in folded hypercubes

This work investigates important properties related to cycles of embedding into the folded hypercube FQn for n ≥ 2. The authors observe that FQn is bipartite if and only if n is odd, and show that the minimum length of odd cycles is n + 1 if n is even. The authors further show that every edge of FQn lies on a cycle of every even length from 4 to 2n; if n is even, every edge of FQn also lies on ...

Edge-fault-tolerant properties of hypercubes and folded hypercubes

Pythagorean powers of hypercubes

For n ∈ N consider the n-dimensional hypercube as equal to the vector space F2 , where F2 is the field of size two. Endow F2 with the Hamming metric, i.e., with the metric induced by the `1 norm when one identifies F2 with {0, 1} ⊆ R. Denote by `2 (F2 ) the n-fold Pythagorean product of F2 , i.e., the space of all x = (x1, . . . , xn) ∈ ∏n j=1 F n 2 , equipped with the metric ∀x, y ∈ n ∏ j=1 F2...

Optimally pebbling hypercubes and powers

We point out that the optimal pebbling number of the n-cube is (4 3) n+O(log n) , and explain how to approximate the optimal pebbling number of the nth cartesian power of any graph in a similar way. Let G be a graph. By a distribution of pebbles on G we mean a function a : V (G) ! Z 0 ; we usually write a(v) as a v , and call a v the number of pebbles on v. A pebbling move on a distribution cha...