Leighton's Theorem: extensions, limitations, and quasitrees

TL;DR

This paper investigates the limitations and extensions of Leighton's Theorem, showing it applies to regular covers by quasitrees but not to non-regular covers or covers by graphs other than trees.

Contribution

The paper clarifies the scope of Leighton's Theorem, establishing its validity for regular quasitree covers and identifying its limitations for non-regular and non-tree covers.

Findings

Leighton's Theorem does not extend to regular covers by graphs other than trees.

The theorem does not hold for non-regular covers by quasitrees, even with a uniform lattice.

It does extend to regular coverings by quasitrees.

Abstract

Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $\hat G$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.