Given a graph and a weight function defined on the vertex set of a graph, the maximum weighted clique (MWC) problem calls for finding the number of vertices with maximum total weight and also any two of vertices are pairwise adjacent. In this paper, an edge based local search algorithm, called ELS, is proposed for the MWC, a well-known combinatorial optimization problem. ELS is a two phased loc...

By using the concepts of ant imetry and n-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. I t is shown tha t tbe number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called an n-clique, is introduced. n...

The weak chromatic number , or clique chromatic number (CCHN) of a graph is the minimum number of colors in a vertex coloring, such that every maximal clique gets at least two colors. The weak chromatic index , or clique chromatic index (CCHI) of a graph is the CCHN of its line graph. Most of the results here are upper bounds for the CCHI, as functions of some other graph parameters, and contra...

The visibility graph V(P ) of a point set P ⊆ R has vertex set P , such that two points v, w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P ). We characterise the 2and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main r...

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [7,11] and claw-free graphs [6,7]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [8]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, ...

Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [8, 12] and claw-free graphs [7, 8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, i...

In [6] we computed the edge Folkman number F (3, 4; 8) = 16. There we used and announced without proof that in any blue-red coloring of the edges of the graph K1 +C5 +C5 + C5 there is either a blue 3-clique or red 4-clique. In this paper we give a detailed proof of this fact.

A weakening of Hadwiger’s Conjecture states that every n-vertex graph with independence number α has a clique minor of size at least n α . Extending ideas of Fox [6], we prove that such a graph has a clique minor with at least n (2−c)α vertices where c > 1/19.2.

Let qmin(G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n: This paper gives some results on the following extremal problem: How large can qmin (G) be if G is a graph of order n; with no complete subgraph of order r + 1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds ...