On the Number of Acyclic Orientations of Complete $k$-Partite Graphs

TL;DR

This paper presents a recurrence-based dynamic programming method to efficiently compute the number of acyclic orientations in complete k-partite graphs, extending to near-complete cases by edge removal.

Contribution

It introduces a novel recurrence relation linking acyclic orientations and Hamiltonian paths, enabling polynomial-time computation for these graph classes.

Findings

Developed a recurrence for acyclic orientations

Implemented a dynamic programming algorithm with n^{O(k)} complexity

Extended the method to graphs close to complete k-partite structure

Abstract

Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time $n^{O(k)}$, where $n$ is the number of vertices in the graph. We prove our result by using a relationship between the number of acyclic orientations and the number of Hamiltonian paths in complete $k$-partite graphs and providing a recurrence for the latter quantity. We give a simple extension of our algorithm to the situation when we are an edge removal away from having a complete $k$-partite graph.