The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

In this paper we consider a branch-and-bound algorithm for the maximum clique problem. We introduce cost based filtering techniques for the so-called candidate set (i.e. a set of nodes that can possibly extend the clique in the current choice point). Doing this, we can reduce the number of choice points visited by a typical factor of 10 – 50. Additionally, we present a taxonomy of upper bounds ...

An edge based local search algorithm, called ELS, is proposed for the maximum clique problem (MCP), a well-known combinatorial optimization problem. ELS is a two phased local search method effectively £nds the near optimal solutions for the MCP. A parameter ’support’ of vertices de£ned in the ELS greatly reduces the more number of random selections among vertices and also the number of iteratio...

A circle graph is the intersection graph of chords drawn in a circle. The best known general upper bound on the chromatic number of circle graphs with clique number k is 50 · 2. We prove a stronger bound of 2k − 1 for graphs in a simpler subclass of circle graphs, so called clean graphs. Based on this result we prove that the chromatic number of every circle graph with clique number at most 3 i...

In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph) which is also called a flag complex, and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersect...

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R2. First, we propose an O(n2 log n) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangle...

We approximately solve, by reduction to Maximum Clique, the graph k-coloring NP-hard problem in a binary weights Hoppeld net special case. This network was used earlier to approximately solve Maximum Clique and some other NP-hard problems by reduction to Maximum Clique. We determine k-coloring approximation performance on random graphs and on one other distribution of \harder" graphs. We compar...

In an earlier paper, we proved (with Chudnovsky and Spirkl) that for all integers κ, ` ≥ 0, every graph with clique number at most κ and sufficiently large chromatic number has an odd hole of length at least `. Here we prove a strengthening, that for all integers κ, ` ≥ 0, every graph with clique number at most κ and sufficiently large chromatic number has a hole of every possible length modulo `.

A family C of sets has the Helly property if any subfamily C′, whose elements are pairwise intersecting, has non-empty intersection. Suppose C is a non-empty family of subsets of a finite set V . The Helly number h(C) of C is the smallest positive integer n such that every subfamily C′ of C with |C′| ≥ n and which intersects n-wise has non-empty intersection. In this paper we consider the famil...

The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. Announced in [40]), the proof uses now an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpo...