On the matroidal path ideals

TL;DR

This paper establishes that fixed-length paths in complete multipartite graphs form matroid bases and explores algebraic properties like Cohen-Macaulayness and depth of their associated path ideals.

Contribution

It introduces the matroidal structure of path sets in complete multipartite graphs and analyzes algebraic properties of their path ideals.

Findings

Paths of fixed length form matroid bases in complete multipartite graphs

Powers of t-path ideals exhibit Cohen-Macaulayness under certain conditions

Depth of t-path ideals is characterized in the context of complete multipartite graphs

Abstract

We prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of $t$-path ideals of a complete multipartite graph.