The minimum forcing number of perfect matchings in the hypercube

TL;DR

This paper proves that the minimum forcing number of perfect matchings in an n-dimensional hypercube is at least 2^{n-2} for all n ≥ 2, confirming a longstanding conjecture using linear algebra techniques.

Contribution

It confirms the conjecture that the forcing number in hypercubes is at least 2^{n-2} for all dimensions, extending previous results for even n to all n.

Findings

Forcing number of perfect matchings in hypercubes is at least 2^{n-2} for all n ≥ 2.

The proof employs simple linear algebra methods.

The conjecture by Pachter and Kim is fully proved.

Abstract

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim [Discrete Math. 190 (1998) 287--294] conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is at least $2^{n-2}$, for all $n \ge 2$. Riddle [Discrete Math. 245 (2002) 283-292] proved this for even $n$. We show that the conjecture holds for all $n \ge 2$. The proof is based on simple linear algebra.