Caterpillar Factorizations of Crowns

Abstract

For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : 0 \leq i \leq n-1, j = i+1, i+2, \ldots, i+k \pmod{n}\}\). 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.