On the lower bound of k-maximal digraphs

On the lower bound of k-maximal digraphs

For a digraph D, let λ(D) be the arc-strong-connectivity of D. For an integer k > 0, a simple digraph Dwith |V (D)| ≥ k + 1 is k-maximal if every subdigraph H of D satisfies λ(H) ≤ k but for adding new arc to D results in a subdigraph H ′ with λ(H ) ≥ k + 1. We prove that if D is a simple k-maximal digraph on n > k + 1 ≥ 2 vertices, then |A(D)| ≥ n 2  + (n − 1)k +  n k + 2   1 + 2k −  k +...

On k-Maximal Strength Digraphs

Let k > 0 be an integer and let D be a simple digraph on n > k vertices. We prove that If |A(D)| > k(2n − k − 1)+ ( n − k 2 )

On the Maximal Degree of the K-Step Obrechkoff’s Method

Note on Maximal Bisection above Tight Lower Bound

In a graph G = (V,E), a bisection (X,Y ) is a partition of V into sets X and Y such that |X| ≤ |Y | ≤ |X|+1. The size of (X,Y ) is the number of edges between X and Y . In the Max Bisection problem we are given a graph G = (V,E) and are required to find a bisection of maximum size. It is not hard to see that ⌈|E|/2⌉ is a tight lower bound on the maximum size of a bisection of G. We study parame...

Lower Bound for the Size of Maximal Nontraceable Graphs

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n) ≥ d3n−2 2 e−2 for n ≥ 20 and g(n) ≤ d3n−2 2 e for n ≥ 54 as well as for n ∈ I = {22, 23, 30, 31, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51}. We show that g(n) = d3n−2 2 e for n ≥ 54 as well as for n ∈ I ∪ {12, 13} and we determine g(n) for n ≤ 9.