# A Variation of the Erd\H{o}s-S\'os Conjecture in Bipartite Graphs

## TL;DR

This paper explores a bipartite graph variation of the Erd ext{"o}s-S ext{"o}s Conjecture, determining maximum sizes of bipartite graphs that exclude certain bipartite trees and characterizing extremal cases.

## Contribution

It extends the Erd ext{"o}s-S ext{"o}s Conjecture to bipartite graphs, identifying maximum sizes and characterizing extremal bipartite graphs avoiding specific tree subgraphs.

## Key findings

- Maximum size of bipartite graphs avoiding all (n,m)-bipartite trees determined
- Maximum size of bipartite graphs avoiding all (k,2)-bipartite trees determined
- Extremal graphs characterized for these cases

## Abstract

The Erd\H{o}s-S\'{o}s Conjecture states that every graph with average degree more than $k-2$ contains all trees of order $k$ as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an $(n,m)$-bipartite graph which does not contain all $(k,l)$-bipartite trees for given integers $n\ge m$ and $k\ge l$. In particular, we determine that the maximum size of an $(n,m)$-bipartite graph which does not contain all $(n,m)$-bipartite trees as subgraphs (or all $(k,2)$-bipartite trees as subgraphs, respectively). Furthermore, all these extremal graphs are characterized.