This paper studies the connectivity of a class of random graphs. An important feature of the class of random graphs is that the edges in the graph are not chosen independently of each other. These graphs are constructed as follows. Start with a complete graph whose edge weights are independently and uniformly distributed in the interval \([0, 1]\). Now have each vertex color the \(k\) edges with smallest weight incident to it green. The subgraph composed of green edges is the object of study. Such graphs can be viewed as the undirected versions of the more well-known \(k\)-out random graphs. The authors derive, for \(k= o(\log n)\), the number of edges in such a graph (with high probability of course). For \(k= 2\), they establish that with high probability the graph is either connected or contains a giant connected component with a scattering of cyclic components. By contrast, in the traditional random graph model, once the degree of each vertex is at least 1, it is connected with high probability. In fact they show that the probability of connectedness for \(k= 2\) is about 0.99. For \(k\geq 3\), such graphs are connected with high probability.