Properties of Generalized Derangement Graphs

TL;DR

This paper studies properties of generalized derangement graphs, characterizing their connectivity and Eulerian nature, and determines key graph parameters for specific cases like the 2-derangement graph when n is an odd prime power.

Contribution

It provides a characterization of connectivity and Eulerian properties of generalized derangement graphs and computes their independence, clique, and chromatic numbers for certain cases.

Findings

Characterized when generalized derangement graphs are connected or Eulerian.

Determined independence, clique, and chromatic numbers for 2-derangement graphs when n is an odd prime power.

Established structural properties of these graphs based on permutation derangements.

Abstract

A permutation sigma in Sn is a k-derangement if for any subset X = {a1, . . ., ak} \subseteq [n], {sigma(a1), . . ., sigma(ak)} is not equal to X. One can form the k-derangement graph on the set of permutations of Sn 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 number of the 2-derangement graph.