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

Abstract

A bipartite graph G is bipancyclic if G has a cycle of length l for every even $4⩽l⩽|V(G)|$. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph $B_n$ is bipancyclic for $n⩾4$ and also show that it is edge-bipancyclic for $n⩾5$. Assume that F is a subset of $E(B_n)$. We prove that $B_n-F$ is bipancyclic, when $n⩾4$ and $|F|⩽n-3$. Since $B_n$ is a $(n-1)$-regular graph, this result is optimal in the worst case.