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arXiv · 2501.05968

Oriented discrepancy of Hamilton cycles in oriented graphs satisfying Ore-type condition

Jiangdong Ai·Qiwen Guo·Gregory Gutin·Yongxin Lan·Qi Shao·Anders Yeo·Yacong Zhou
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Abstract

Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following extension of Dirac's theorem: If $D$ is an oriented graph on $n \ge 3$ vertices with minimum degree $\delta (D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $\delta(D)$ arcs in the same direction. This conjecture was proved by Freschi and Lo (2024) who posed an open problem to extend their result to an Ore-type condition. We propose two conjectures for such extensions and prove results which provide support to the conjectures.

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Jiangdong Ai, Qiwen Guo, Gregory Gutin, Yongxin Lan, Qi Shao, Anders Yeo, Yacong Zhou. 2025-01-10. Oriented discrepancy of Hamilton cycles in oriented graphs satisfying Ore-type condition. https://arxiv.org/abs/2501.05968

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