  We introduce the notion of a graph derangement, which naturally interpolates
between perfect matchings and Hamiltonian cycles. We give a necessary and
sufficient condition for the existence of graph derangements on a locally
finite graph. This result was first proved by W.T. Tutte in 1953 by applying
some deeper results on digraphs. We give a new, simple proof which amounts to a
reduction to the (Menger-Egervary-Konig-)Hall(-Hall) Theorem on transversals of
set systems. Finally, we consider the problem of classifying all cycle types of
graph derangements on m x n checkerboard graphs. Our presentation does not
assume any prior knowledge in graph theory or combinatorics: all definitions
and proofs of needed theorems are given.
