The existence of near-Rosa and hooked near-Rosa sequences Original Research Article

Let m,n be positive integers, m⩽n. A near-Rosa sequence of order n and defect m is a sequence R=(r1,r2,…,r2n−1) of integers ri∈{1,2,…,m−1,m+1,…n} which satisfy the following conditions: 1. For every k∈{1,2,…,m−1,m+1,…,n} there are exactly two elements ri,rj∈R, such that ri=rj=k, 2. If ri=rj=k, j>i, then j−i=k, 3. rn=0. 4. A hooked near-Rosa sequence of order n and defect m is a sequence HR=(r1,r2,…,r2n) of positive integers ri∈{1,2,…,m−1,m+1,…,n} satisfying conditions (1), (2) and the condition: 5. rn+1=r2n−1=0. We show that the necessary conditions for the existence of near-Rosa and hooked near-Rosa sequences are also sufficient.