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.

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic 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 monochromatic graph isomorphic to H. Let f_{k}(n,H) be the smallest number t such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H-decomposition with at most t parts. Here we study the function f_{k}(n,K_r) for k≥ 2 and r≥ 3.

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic 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 monochromatic graph isomorphic to H. Let f_{k}(n,H) be the smallest number t such that any k-edge-colored graph G of order n, admits a monochromatic H-decomposition with at most t parts. Here we study the function f_{k}(n,K_r) for k ≥2 and r≥ 3.

An edge-coloured path is rainbow if its edges have distinct colours. The rainbow connection number of a connected graph G, denoted by rc(G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by a rainbow path. The function rc(G) was first introduced by Chartrand et al.~[Math.~Bohem., 133(1) (2008), pp.85-98], and has since attracted considerable interest. In this paper, we introduce two extensions of the rainbow connection number to hypergraphs. We study these two extensions of the rainbow connection number in minimally connected hypergraphs, hypergraph cycles and complete multipartite hypergraphs.

Let k be a positive integer and G be a k-connected graph. An edge-coloured path is rainbow if its edges have distinct colours. The rainbow k-connection number of G, denoted by rc_k(G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function rc_k(G) was first introduced by Chartrand, Johns, McKeon, and Zhang in 2009, and has since attracted considerable interest. In this paper, we consider a version of the function rc_k(G) which involves vertex-colourings. A vertex-coloured path is vertex-rainbow if its internal vertices have distinct colours. The rainbow vertex k-connection number of G, denoted by rvc_k(G), is the minimum number of colours required to colour the vertices of G so that any two vertices of G are connected by k internally vertex-disjoint vertex-rainbow paths. We shall study the function rvc_k(G) when G is a cycle, a wheel, and a complete multipartite graph. We also construct graphs G where rc_k(G) is much larger than rvc_k(G) and vice versa so that we cannot in general bound one of rc_k(G) and rvc_k(G) in terms of the other.

Let k be a positive integer and G be a k-connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k-connection number rc_k(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rc_k(G) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function rc_k(G) has since been studied by numerous researchers. An analogue of the function rc_k(G) involving vertex colourings, the rainbow vertex k-connection number rvc_k(G), was subsequently introduced. In this paper, we introduce a version which involves total colourings. A total-coloured path is total-rainbow if its edges and internal vertices have distinct colours. The total rainbow k-connection number of G, denoted by trc_k(G), is the minimum number of colours required to colour the edges and vertices of $G$, so that any two vertices of $G$ are connected by $k$ internally vertex-disjoint total-rainbow paths. We study the function trc_k(G) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions rc_k(G), rvc_k(G), and trc_k(G), by considering how close and how far apart trc_k(G) can be from rc_k(G) and rvc_k(G).