Multipath matroids, digraph colourings, and the Tutte polynomial

TL;DR

This paper characterizes MP-digraphs where multipaths form a matroid, relates their Tutte polynomial to digraph colorings, and connects multipath cohomology decategorification to Tutte polynomial specializations.

Contribution

It introduces the class of MP-digraphs, establishes their Tutte polynomial relationships, and links multipath cohomology to Tutte polynomial specializations.

Findings

Characterization of MP-digraphs where multipaths form a matroid

Relation between Tutte polynomial and digraph colorings for MP-digraphs

Decategorification of multipath cohomology yields Tutte polynomial specialization

Abstract

We characterise the digraphs for which the multipaths, that is disjoint unions of directed paths, yield a matroid. For such graphs, called MP-digraphs, we prove that the Tutte polynomial of the multipath matroid is related to counting certain digraph colourings. Finally, we prove that, for MP-forests, the decategorification of the multipath cohomology yields a specialisation of the Tutte polynomial.