Sweeping graphs with large clique number

Searching Graphs with Large Clique Number

Colouring perfect graphs with bounded clique number

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver [9] from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a “combinatorial” polynomial-time algorithm that yields an o...

Dense graphs with small clique number

We consider the structure of Kr-free graphs with large minimum degree, and show that such graphs with minimum degree δ > (2r − 5)n/(2r − 3) are homomorphic to the join Kr−3 ∨ H where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that Kr-free graphs with such minimum degree have chromatic number at most r+ 1. We also consider th...

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

Let Z be the set of all integers and M a set of positive integers. The distance graph G(Z,M) generated by M is the graph with vertex set Z and in which i and j are adjacent whenever |i − j| ∈ M . Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council, R. O. C., under grant NSC892115-M-110-012.

Minimal circular-imperfect graphs of large clique number and large independence number

This paper constructs some classes of minimal circular-imperfect graphs. In particular, it is proved that there are minimal circularimperfect graphs whose independence number and clique number are both arbitrarily large.