A classical result of Hajnal and Szemerédi, when translated to a complementary form, states that with sufficient minimum degree, a graph will contain disjoint large cliques. We conjecture a generalization of this result from cliques to cycles with many chords and prove this conjecture in several cases.

A coloring of the edges of Kn is constructed such that every copy of K4 has at least three colors on its edges. As n → ∞, the number of colors used is eO( √ log n ). This improves upon the previous probabilistic bound of O( √ n ) due to Erdős and Gyárfás.

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number cliques any size k contained in geometric inhomogeneous random graph: scale-free network model containing geometry. The interplay between scale-freeness geometry ensures that connections are likely form either high-degree vertices, or close by vertices. At same...