Bipartite Tur\'an number of paths and other trees

TL;DR

This paper determines the maximum number of edges in bipartite graphs with a given longest path length, extending previous results and exploring generalizations to other trees.

Contribution

It provides an exact solution for bipartite graphs with arbitrary bipartition sizes and longest path lengths, generalizing earlier special cases.

Findings

Exact maximum edge count for bipartite graphs with specified longest path

Extension of results to graphs with unequal bipartition sizes

Discussion on generalizations to other tree structures

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.