For positive integers k <= n, the crown C(n,k) is the graph with vertex set {a(0), a(1),..., a(n-1), b(0), b(1),..., b(n-i)} and edge set {a(i)b(j) : 0 <= i <= 1, j = i + 1, i + 2,..., i + k ( mod n)}. A caterpillar is a tree of order at least three which contains a path such that each vertex not on the path is adjacent to a vertex on the path. Being a connected bipartite graph, a caterpillar is balanced if the two parts of the bipartition of its vertices have equal size; otherwise, it is unbalanced. In this paper we obtain the necessary and sufficient condition for balanced-caterpillar factorization of crowns. The Criterion for unbalanced-caterpillar factorization of crowns is open. We also obtain the necessary and sufficient condition for directed caterpillar factorization of symmetric crowns.
