In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − →rc(T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of kth diameter 2 has rainbow k-co...

A path in an edge-colored graph is called a rainbow path if the edges on it have distinct colors. For k ≥ 1, the rainbow-k-connectivity of a graph G, denoted rck(G), is the minimum number of colors required to color the edges of G in such a way that every two distinct vertices are connected by at least k internally vertex-disjoint rainbow paths. In this paper, we study rainbow-k-connectivity in...

A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least k/2 . A properly edge-coloured K4 has no such matching, which motivates ...

This paper studies the computational complexity of the Edge Packing problem and the Vertex Packing problem. The edge packing problem (denoted by EDS ) and the vertex packing problem (denoted by DS ) are linear programming duals of the edge dominating set problem and the dominating set problem respectively. It is shown that these two problems are equivalent to the set packing problem with respec...

In this paper, we discuss the approximability of the capacitated b-edge dominating set problem, which generalizes the edge dominating set problem by introducing capacities and demands on the edges. We present an approximation algorithm for this problem and show that it achieves a factor of 8/3 for general graphs and a factor of 2 for bipartite graphs. Moreover, we discuss the relationships of t...

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such...

An edge-colored graph G is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ ( n−2 2 ) + 2, and rc(G) ≤ 4 if |E(G)| ≥ ( n−3 2 ) + 3. These bounds...

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colours needed for a rainbow edge (vertex) colouring of G. In this paper we propose a very simple approach to studying rainbow connectivity in...

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such ...

In this paper we consider properly edge-colored graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A matching is called rainbow if its edges have different colors. The minimum degree of a graph is denoted by δ(G). We show that properly edge colored graphs G with |V (G)| ≥ 4δ(G) − 3 have rainbow matchings of size δ(G), this gives the best know...