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Main Authors:	Guo, Q., Gutin, G., Lan, Y., Shao, Q., Yeo, A., Zhou, Y.
Format:	Preprint
Published:	2026
Subjects:	Combinatorics
Online Access:	https://arxiv.org/abs/2602.10713
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author	Guo, Q. Gutin, G. Lan, Y. Shao, Q. Yeo, A. Zhou, Y.
author_facet	Guo, Q. Gutin, G. Lan, Y. Shao, Q. Yeo, A. Zhou, Y.
contents	Gishboliner, Krivelevich, and Michaeli (2023) conjectured the following generalization of Dirac's theorem: If the minimum degree $δ$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$, then $G$ has a Hamilton oriented cycle with at least $δ$ 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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2602_10713
institution	arXiv
publishDate	2026
record_format	arxiv
spellingShingle	Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments Guo, Q. Gutin, G. Lan, Y. Shao, Q. Yeo, A. Zhou, Y. Combinatorics Gishboliner, Krivelevich, and Michaeli (2023) conjectured the following generalization of Dirac's theorem: If the minimum degree $δ$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$, then $G$ has a Hamilton oriented cycle with at least $δ$ 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.
title	Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments
topic	Combinatorics
url	https://arxiv.org/abs/2602.10713

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