Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique K such that for every vertex v ...

A hereditary class of graphs is -bounded if there is a function f such that every graph G in the class has chromatic number at most f(!(G)), where !(G) is the clique number of G; and the class is polynomially bounded ...

A hole in a graph G is an induced cycle of length at least four, and a k-multihole in G is the union of k pairwise disjoint and nonneighbouring holes. It is well known that if G does not contain any holes then its chromatic ...

The Erdős-Hajnal conjecture asserts that for every graph H there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|c. In this ...

A graph is H-free if it has no induced subgraph isomorphic to H, and |G| denotes the number of vertices of G. A conjecture of Conlon, Sudakov and the second author asserts that: - For every graph H, there exists ∈ > 0 ...

We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in C as an induced ...

The Erdős-Hajnal conjecture says that for every graph H there exists c > 0 such that max(α(G), w(G)) ≥ nc for every H-free graph G with n vertices, and this is still open when H = C5. Until now the best bound known on ...

We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vušković.