Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4 /spl les/l/spl les/|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length I of G. In this paper, we show that the bubble-sort graph B/sub n/ is bipancyclic for n/spl ges/ 4, and also show that it is edge-bipancyclic for n/spl ges/5. To obtain this results, we also prove that we can construct a hamiltonian cycle of B/sub n/ that contains given two nonadjacent edges. Assume that F is the subset of E(B/sub n/). We prove that B/sub n /-F is bipancyclic whenever n /spl ges/4 and |F|/spl les/ n-3. Since B/sub n/ is a (n-1)-regular graph, this result is optimal in the worst case.