On graphs with equal coprime index and clique number

On Generalized Coprime Graphs

Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.

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...

Searching Graphs with Large Clique Number

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...

On the Spectral Moment of Graphs with given Clique Number

Let Ln,t be the set of all n-vertex connected graphs with clique number t (2 ≤ t ≤ n). For nvertex connected graphs with given clique number, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The first ∑⌊(n−t−1)/3⌋ i=1 (n − t − 3i) + 1 graphs with 3 ≤ t ≤ n − 4, and the last few graphs, in the Sorder, among Ln,t are characterized. In addition, all graphs in Ln,n ∪...