Minimal degree and \((k,m)\)-pancyclic ordered graphs (Q2484156)

Given positive integers \(k\leq m\leq n\), a graph \(G\) of order \(n\) is \((k,m)\)-pancyclic ordered if for any set of \(k\) vertices of \(G\) and any integer \(r\) with \(m\leq r\leq n\), there is a cycle of length \(r\) encountering the \(k\) vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order \(n\) is \((k,m)\)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.