Graph Derangements

TL;DR

This paper introduces graph derangements, a concept bridging perfect matchings and Hamiltonian cycles, providing a new simple proof for their existence criteria and exploring their cycle types on checkerboard graphs.

Contribution

It presents a new, simplified proof for the existence of graph derangements and explores their cycle structures on checkerboard graphs, extending classical results.

Findings

Necessary and sufficient condition for graph derangements on locally finite graphs.

A new proof reduces the problem to the Hall theorem on set system transversals.

Classification of cycle types of graph derangements on checkerboard graphs.

Abstract

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.