Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments

TL;DR

This paper investigates maximizing forward arcs in Hamiltonian cycles and paths within generalized tournament structures, providing characterizations and polynomial algorithms for certain classes, advancing understanding of directed graph Hamiltonicity.

Contribution

It offers new characterizations and polynomial algorithms for maximizing forward arcs in Hamiltonian cycles and paths in specific generalized tournament classes.

Findings

Characterizations for maximum forward arcs in semicomplete multipartite digraphs

Polynomial-time algorithms derived from these characterizations

NP-hardness results for other generalized tournament classes

Abstract

Gishboliner, Krivelevich, and Michaeli (2023) conjectured the following generalization of Dirac's theorem: If the minimum degree $\delta$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$, then $G$ has a Hamilton oriented cycle with at least $\delta$ forward arcs. Freschi and Lo (2024) proved this conjecture. In this paper, we study the problem of maximizing the number of forward arcs in Hamilton oriented cycles/paths in generalizations of tournaments. We obtain characterizations for the maximum number of forward arcs in semicomplete multipartite digraphs and locally semicomplete digraphs. These characterizations lead to polynomial-time algorithms. Note that the above problems are NP-hard for some other generalizations of tournaments even though the Hamilton cycle problem is polynomial-time solvable for these digraph classes.