Teresa Sousa

Total rainbow k-connection in graphs

Abstract 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 r c k ( G ) of G . An edge-coloured path is rainbow if its edges have distinct colours. Then, r c 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 r c k ( G ) has since been studied by numerous researchers. An analogue of the function r c k ( G ) involving vertex colourings, the rainbow vertex k -connection number r v c 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 t r c 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 t r c k ( G ) when G is a cycle, a wheel, and a complete multipartite graph. We also compare the functions r c k ( G ) , r v c k ( G ) , and t r c k ( G ) , by considering how close and how far apart t r c k ( G ) can be from r c k ( G ) and r v c k ( G ) .

Monochromatic Kr-Decompositions of Graphs

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 be the smallest number ϕ 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 ϕ parts. Here, we study the function for and .

Decompositions of Graphs into Fans and Single Edges

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 φ(n,H) be the smallest number ϕ such that any graph G of order n admits an H-decomposition with at most ϕ parts. Pikhurko and Sousa conjectured that φ(n,H)= ex (n,H) for χ(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 verified by Ozkahya and Person for all edge-critical graphs H. In this article, the conjecture is verified for the k-fan graph. The k-fan graph, denoted by Fk, is the graph on 2k+1 vertices consisting of k triangles that intersect in exactly one common vertex called the center of the k-fan.

Monochromatic Clique Decompositions of Graphs

Abstract Let G be a graph whose edges are colored with k colors, and H=(H1,⋯,Hk) be a k‐tuple of graphs. 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 copy of Hi in color i, for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order‐n graph and every k‐edge‐coloring, there is a monochromatic H‐decomposition with at most ϕ elements. Extending the previous results of Liu and Sousa [Monochromatic Kr‐decompositions of graphs, J Graph Theory 76 (2014), 89–100], we solve this problem when each graph in H is a clique and n≥n0(H) is sufficiently large.

Rainbow connection for some families of hypergraphs

Abstract An edge-coloured path in a graph is rainbow if its edges have distinct colours. The rainbow connection number of a connected graph G , denoted by r c ( 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 r c ( G ) was first introduced by Chartrand et al. (2008), 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.

Monochromatic K r -Decompositions of Graphs

Given graphs G and H, and a colouring of the edges of G with k colours, 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 k(n;H) be the smallest number such that any graph G of order n and any colouring of its edges with k colours, admits a monochromatic Hdecomposition with at most parts. Here we study the function k(n;Kr) for k 2 and r 3. This work was partially supported by FCT - Funda c~ ao para a Ci^ encia e a Tecnologia through Projects PTDC/MAT/113207/2009 and PEst-OE/MAT/UI0297/2011 (CMA).

Friendship decompositions of graphs

We investigate the maximum number of elements in an optimal t-friendship decomposition of graphs of order n. Asymptotic results will be obtained for all fixed t>=4 and for t=2,3 exact results will be derived.

Minimum H-Decompositions of Graphs and Its Ramsey Version: A Survey

The subject of H-decompositions of graphs was first introduced by Erdős, Goodman and Posa in 1966. Given 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 ϕ(n, H) be the smallest number ϕ, such that, any graph G with n vertices admits an H-decomposition with at most ϕ parts. The exact computation of ϕ(n, H) for an arbitrary H is still an open problem. In this paper we will survey recent results about H-decompositions of graphs and we will also introduce its Ramsey or coloured version together with recent results on this problem.