Teresa Maria Jerónimo Sousa

Given two graphs G and H, an H-decomposition of G is a partition of the
edge set of G such that each part is either a single edge or forms a graph
isomorphic to H. Let f(n;H) be the smallest number  such that any graph
G of order n admits an H-decomposition with at most f(n;H)  parts. Pikhurko and
Sousa conjectured that f(n;H) = ex(n;H) for (H)  3 and all sufficiently
large n, where ex(n;H) denotes the maximum number of edges in a graph on n vertices not containing H as a subgraph. Their conjecture has been veried by
Ozkahya and Person for all edge-critical graphs H. In this article, the conjecture
is veried for the k-fan graph. The k-fan graph, denoted by F_k, is the graph on
2k + 1 vertices consisting of k triangles which intersect in exactly one common
vertex called the centre of the k-fan.

Let G be a simple graph. A clique is a subgraph of G isomorphic to a complete graph, and a t-friendship graph is a graph consisting of t edge-disjoint cliques sharing one vertex. A t-friendship decomposition of G for a fixed value of t is a set F of edge-disjoint t-friendship subgraphs of G such that every edge of G belongs to exactly one element of F. It is proved that any graph of order n admits a t-friendship decomposition with at most n2/4t+n/4t+n elements for all fixed t≥2. For t=2,3 exact bounds are given. For t=2 the bound is ⌈n2/8⌉ for n even and (n2−1)/8 for n odd; for t=3 the bound is ⌈n2/12⌉ for n even and ⌈(n2−1)/12⌉ for n odd.

A friendship graph is a graph consisting of cliques sharing a common vertex. In this paper we investigate the maximum number of elements in an optimal friendship decomposition of graphs of order n. We obtain upper and lower bounds for this number. These bounds relate this problem with the classical Ramsey numbers.