A new matroid lift construction and an application to group-labeled graphs

TL;DR

This paper generalizes a classical matroid lift construction to higher ranks and applies it to group-labeled graphs, characterizing when such lifts exist for abelian groups based on finite field structures.

Contribution

It introduces a new method to construct rank-k matroid lifts from circuit sets and applies this to characterize group-labeling lifts in graphs.

Findings

Generalized Brylawski's matroid lift construction to rank-k.

Proposed a conjecture that all matroid lifts can be obtained via this method.

Characterized when group-labeled graph lifts exist for abelian groups.

Abstract

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by showing how to construct a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. We conjecture that every lift of $M$ arises via this construction. We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with edges labeled by a group, Zaslavsky's lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the group-labeling; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that are balanced with respect to the group-labeling. For $k \ge 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field.