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arXiv · 2602.10713

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

Q. Guo·G. Gutin·Y. Lan·Q. Shao·A. Yeo·Y. Zhou
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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.

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Q. Guo, G. Gutin, Y. Lan, Q. Shao, A. Yeo, Y. Zhou. 2026-02-11. Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments. https://arxiv.org/abs/2602.10713

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