Isomorphic Path Decompositions of Crowns

Abstract

For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n\}\) and edge set \(\{a_ib_j: 1 \leq i \leq n, j = i,i+1,\ldots, i+k-1 \pmod{n}\}\). In this paper, we give a necessary and sufficient condition for the existence of a \(P_1\) decomposition of \(C_{n,k}\).