The chromatic polynomial P (G,λ) gives the number of λ-colourings of a graph. If P (G,λ) = P (H1, λ)P (H2, λ)/P (Kr , λ), then the graph G is said to have a chromatic factorisation with chromatic factors H1 and H2. It is known that the chromatic polynomial of any clique-separable graph has a chromatic factorisation. In this paper we construct an infinite family of graphs that have chromatic fac...

In this paper we study the edge-thickness and the clique-thickness of a graph. The edge-thickness of a graph is defined as the thickness of the family of edges. The clique-thickness of a graph is defined as the thickness of the family of maximal cliques. Edges and maximal cliques of a graph are both considered as a collection of subsets of the vertex set. On one hand, we introduce a new paramet...

We define a multivariate polynomial that generalizes several interlace polynomials defined by Arratia, Bollobas and Sorkin on the one hand, and Aigner and van der Holst on the other. We follow the route traced by Sokal, who defined a multivariate generalization of Tutte’s polynomial. We also show that bounded portions of our interlace polynomial can be evaluated in polynomial time for graphs of...

A clique in an undirected graph G= (V, E) is a subset V ' ⊆ V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is NP-Complete. We have succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination. For a graph o...

Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An nvertex random graph has each edge included with probability 1 2 , and a clique is a completely conn...

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

We study the size of the largest clique ω(G(n, α)) in a random graphG(n, α) on n vertices which has power-law degree distribution with exponent α. We show that for ‘flat’ degree sequences with α > 2 whp the largest clique in G(n, α) is of a constant size, while for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm wh...

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of m labels is chosen independently with probability p by each ...

In a paper published in Journal of Combinatorial Theory, Series B (1986), Monma and Wei propose an extensive study of the intersection graph of paths on a tree. They explore this notion by varying the notion of intersection: the paths are respectively considered to be the sets of their vertices and the sets of their edges, and the trees may or may not be directed. Their main results are a chara...

For the Maximum Clique problem we propose a neural approximation algorithm that can be implemented on Field Programmable Gate Arrays (FPGA). The algorithm builds a sequence of discrete Hoppeld networks that, in polynomial time, converge to a state representing a clique for a given graph. Some experiments made on the DIMACS benchmark show that the approximated solutions found are satisfactory. M...