Rainbow connection number of Cm o Pn and Cm o Cn

A lower bound for the crossing number of Cm × Cn

Lower Bounds for the Exponential Domination Number of Cm × Cn

A vertex v in an exponential dominating set assigns weight ( 1 2 )dist(v,u) to vertex u. An exponential dominating set of a graph G is a subset of V (G) such that every vertex in V (G) has been assigned a sum weight of at least 1. In this paper the exponential dominating number for the graph Cm × Cn, denoted by γe(Cm × Cn) is discussed. Anderson et. al. [1] proved that mn 15.875 ≤ γe(Cm × Cn) ≤...

On Exponential Domination of Cm × Cn

An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connecti...

Intersection of Curves and Crossing Number of Cm x Cn on Surfaces

Let (K 1 , K 2) be two families of closed curves on a surface S, such that |K 1 | = m, |K 2 | = n, m 0 ≤ m ≤ n, each curve in K 1 intersects each curve in K 2 , and no point of S is covered three times. When S is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in K 1 ∪ K 2 is at least 10mn/9, 12mn/11, and mn + 10 −13 m 2 , respectively. More...

Extraction on Connection Machine CM - 5

(Summary of Results) Abstract In this paper, we present a parallel implementation of an image feature extraction task on Connection Machine CM-5. We show that, given a 2048 2048 grey level image as input, the extraction of image features, which includes edge detection, thinning, linking, and linear approximation , can be performed in less than 1.2 seconds on a partition of CM-5 having 512 proce...