Properties of generalized derangement graphs

Abstract

A permutation on $n$ elements is called a $k$-derangement ($k\le n$) if no $k$-element subset is mapped to itself. One can form the $k$-derangement graph on the set of all permutations on $n$ elements by connecting two permutations $\sigma$ and $\tau$ if $\sigma {\tau }^{-1}$ is a $k$-derangement. We characterize when such a graph is connected or Eulerian. For $n$ an odd prime power, we determine the independence, clique and chromatic numbers of the 2-derangement graph.