Enumeration of paths and cycles and e-coefficients of incomparability graphs

TL;DR

This paper establishes a relationship between Hamiltonian paths and cycle covers in the complement of acyclic digraphs, and applies this to expand the chromatic symmetric function of incomparability graphs, revealing new combinatorial insights.

Contribution

It introduces a novel equality between Hamiltonian paths and cycle covers in certain digraphs and provides a new expansion of chromatic symmetric functions for incomparability graphs.

Findings

Number of Hamiltonian paths equals cycle covers in the complement of acyclic digraphs

New expansion of chromatic symmetric function in elementary symmetric functions

Bijections involving acyclic orientations derived from the expansion

Abstract

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations.