The clique-transversal number τc(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number αc(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph co...

Let B and R be two simple graphs with vertex set V , and let G(B,R) be the simple graph with vertex set V , in which two vertices are adjacent if they are adjacent in at least one of B and R. For X ⊆ V , we denote by B|X the subgraph of B induced by X; let R|X and G(B,R)|X be defined similarly. A clique in a graph is a set of pairwise adjacent vertices. A subset U ⊆ V is obedient if U is the un...

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. We have recently shown that any greedy clique decomposition of a graph of order n has at most-~ cliques. A greedy max-clique decomposition is a particular kind of greedy clique decomposition where maximum cliques are removed, instead of just maximal ones. In this pa...

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g  and  [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number a...

This paper proposes and investigates a framework for clique gossip protocols. As complete subnetworks, the existence of cliques is ubiquitous in various social, computer, and engineering networks. By clique gossiping, nodes interact with each other along a sequence of cliques. Clique-gossip protocols are defined as arbitrary linear node interactions where node states are vectors evolving as lin...

Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα ≥ n, Hadwiger’s conjecture implies that G has a clique minor of size n α . In this paper we prove that this is true for connected claw-free graphs with α ≥ 3. We also show that this result is tight by pro...

The following theorem is due to Dilworth [12]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques). In this article we formalize an elegant proof of the above theorem for finite posets by Perles [17]. The result is then used in proving the case of infinite posets following the original pro...