Oriented discrepancy of Hamilton cycles and paths in digraphs

2025-01-10

AI Summary

This study addresses the forward-arc maximization problem for Hamilton cycles in directed graphs, focusing on digraphs with minimum semidegree at least half the number of vertices. Methodologically, it systematically characterizes extremal structures achieving the maximum number of forward arcs in Hamilton cycles and paths within semicomplete multipartite and local semicomplete digraphs. It proposes a generalized Ore-type sufficient condition for forward-arc maximality and partially verifies this conjecture. Crucially, it establishes the first necessary and sufficient structural characterizations for the maximum forward-arc count in these two classes, and—based on these characterizations—designs polynomial-time algorithms for both decision and construction. The contributions extend classical Hamiltonicity theory into the domain of directional optimization, providing novel paradigms for extremal structural analysis and combinatorial optimization in directed graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 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. 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 some results which provide support to the conjectures. For forward arc maximization on Hamilton oriented cycles and paths in semicomplete multipartite digraphs and locally semicomplete digraphs, we obtain characterizations which lead to polynomial-time algorithms.

Problem

• Directed Graphs
• Hamiltonian Cycle
• Optimization Algorithm

Innovation

• Hamiltonian cycle
• directed graphs
• computational efficiency