Journal of Algorithms and Computation
Volume:44 Issue: 1, Dec 2013

In this paper we construct antimagic labelings of the regular complete multipartite graphs and we also extend the construction to some families of regular graphs.

Let G be a graph with p vertices and q edges and let A = n 0, 1, 2, . . . ,  q 2  o . A vertex labeling f : V (G) → A induces an edge labeling f ∗ defined by f ∗ (uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 1, 2, 3, . . . , q. In this paper, we prove that TOPˆ n, TOˆ2Pn, TOCˆ n, TOC˜ n are vertex equitable graphs.The formula is not displayed correctly!

Avila and Molina [8] introduced the notion of generalized weak structures which naturally generalize minimal structures, generalized topologies and weak structures and the structures α(g),π(g),σ(g) and β(g). This work is a further investigation of generalized weak structures due to Avila and Molina. Further we introduce the structures ro(g) and rc(g) and study the properties of the structures ro(g), rc(g), and also further properties of α(g),π(g),σ(g) and β(g) due to [8].the formula is not displayed correctly!

Covering arrays of strength two have been widely studied as combinatorial models of software interaction test suites for pairwise testing. While numerous algorithmic techniques have been developed for the generation of covering arrays with few columns (factors), the construction of covering arrays with many factors and few tests by these techniques is problematic. Random generation techniques can overcome these computational difficulties, but for strength two do not appear to yield a number of tests that is competitive with the fewest known.

A compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. Directed Hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. Sets of extrema of Birkhoff polyhedra are mapped to tours neither in a graph nor its complement and these sets are embedded into disjoint orthogonal spaces as the solution set of a compact formulation. An orthogonal projection of its solution set into the subspace spanned by the Birkhoff polytope is the convex hull of all tours neither in a graph nor its complement. It’s suggested that these techniques might be adaptable for application to linear programming models of network and path problems.