A biclique partition of a graph G is an edge coloring of G such that the edge subgraph formed by the edges of any given color is a complete bipartite graph. A claw is a graph isomorphic to a K1,a for some a. A siamese claw is a graph isomorphic to K2,a for some a. We find necessary and sufficient conditions for Kn to be partitioned into claws K1,a1,…,Kl,al where l=n−1 or l=n. Let Xi(i=1,…,m) be a collection of two-element subsets of {1,…,n}. We study the problem of finding subsets Yi(i=1,…,m) of {1,…,n} such that the siamese claws {(a,b): a∈Xi, b∈Yi} (i=1,…,m) partition the edge set of Kn. It is proved that such Yi's exist if and only if there is a perfect matching in a graph G̃ associated with the graph G, the graph whose edges are the Xi's.