Finding Unavoidable Colorful Patterns in Multicolored Graphs

Multicolored Trees in Random Graphs

Unavoidable patterns

Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every 2 > 0 and positive integer k there is an n(k, 2) such that every 2-edge-coloring of the complete graph of order n ≥ n(k, 2) which has at least 2n2 ) edges in each color contains a member of Fk. This ...

Unavoidable Stars in 3-Graphs

Suppose .F is a collection of 3-subsets of (1,2,..., n). The problem of determining the least integer f(n, k) with the property that if l,FT/ >f(n, k) then .7 contains a k-star (i.e., k 3-sets such that the intersection of any pair of them consists of exactly the same element) is studied. It is proved that, for k odd, f(n, k) = k(k 1) n + P(k-‘) and, for k even,f(n, k) = k(k 3/2) n + F(n + k’).

On unavoidable graphs

How many edges can be in a graph which is forced to be contained in every graph on n verti-ces and e edges? 1n this paper we obtain bounds which are in many cases asymptotically best possible. A well-known theorem of Turán [10,11] asserts that every graph on n vertices and e edges must contain a complete subgraph on m vertices if e~2(m 1)(n-rL)+(2) where r satisfies r-n (mod m-1) and In this pa...

Finding Cross Genome Patterns in Annotation Graphs

Annotation graph datasets are a natural representation of scientific knowledge. They are common in the life sciences where concepts such as genes and proteins are annotated with controlled vocabulary terms from ontologies. Scientists are interested in analyzing or mining these annotations, in synergy with the literature, to discover patterns. Further, annotated datasets provide an avenue for sc...