META-HEURISTIC ALGORITHMS FOR MINIMIZING THE NUMBER OF CROSSING OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS

On the Orchard crossing number of complete bipartite graphs

Let G be an abstract graph. Motivated by the Orchard relation, introduced in [3, 4], we have defined the Orchard crossing number of G [5], in a similar way to the well-known rectilinear crossing number of an abstract graph G (denoted by cr(G), see [1, 8]). A general reference for crossing numbers can be [6]. The Orchard crossing number is interesting for several reasons. First, it is based on t...

Mixed cycle-E-super magic decomposition of complete bipartite graphs

An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ΣνεV(H) f(v) +  ΣeεE(H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥ ...

Mixed cycle-E-super magic decomposition of complete bipartite graphs

An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ∑νεV (H) f(v) + ∑νεE (H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥...

Monotonicity for Complete Graphs and Symmetric Complete Bipartite Graphs

Given a graph G, let fk be the number of forests of cardinality k in G. Then the sequence (fk) has been conjectured to be unimodal for any graph G. In this paper we confirm this conjecture for Kn and Kn,n by showing that the sequence for Kn is strictly increasing (when n ≥ 4) and the sequence for Kn,n is strictly increasing except for the very last term. As a corollary we also confirm the conje...

Heuristic to Reduce the Complexity of Complete Bipartite Graphs To . . .

On the Orchard Crossing Number of the Complete Bipartite Graphs Kn,n

We compute the Orchard crossing number, which is defined in a similar way to the rectilinear crossing number, for the complete bipartite graphs Kn,n.