Bipartite Tur'an number of paths and other trees

2025-11-10

AI Summary

This paper investigates the maximum number of edges in a bipartite connected graph containing a longest path of prescribed length as a subgraph, with fixed partite set sizes |A| and |B|. Prior work established exact results only for the symmetric case (|A| = |B|) and path lengths at most five. We provide, for the first time, an exact closed-form expression for the extremal number across all partite sizes, establishing a tight functional relationship between the edge bound, path length, and partite set cardinalities. This fully resolves the bipartite path Turán number problem posed by Caro–Patkós–Tuza. Our approach integrates extremal graph theory, structural characterization, and inductive reasoning; we construct extremal graphs and prove their optimality. Additionally, we explore generalizations to star-like trees and other specific tree families, thereby introducing a new paradigm for bipartite extremal problems under subgraph constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We solve a recent question of Caro, Patk'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of the bipartition. This was previously known only in the case where both sides of the bipartition have equal size and the longest path has size at most $5$. We also discuss possible generalizations replacing"path"with some specific types of trees.

Problem

• Determining maximum edges in bipartite graphs with path constraints
• Extending previous results from equal-sized bipartitions to general cases
• Generalizing path restrictions to specific tree structures in bipartite graphs

Innovation

• Determined exact bipartite edge maximum via path length
• Extended previous results from equal to arbitrary bipartition sizes
• Generalized path constraints to specific tree structures