Vertex Covers by Edge Disjoint Cliques

Dedicated to the memory of Paul Erdős

Let H be a simple graph having no isolated vertices. An (H,k)-vertex-cover of a simple graph G = (V,E) is a collection of subgraphs of G satisfying

1.  , for all i = 1, ..., r,

2.  ,

3.  , for all , and

4.  each is in at most k of the . We consider the existence of such vertex covers when H is a complete graph, , in the context of extremal and random graphs.