# Terminal-Pairability in Complete Bipartite Graphs

## TL;DR

This paper studies the terminal-pairability problem in complete bipartite graphs, providing improved bounds and sharp results on demand graph properties to ensure pairability.

## Contribution

It offers new bounds and a sharp theorem for demand graphs in bipartite terminal-pairability problems, advancing understanding of this graph theory challenge.

## Key findings

- Improved lower bound on maximum degree for demand graphs to be terminal-pairable.
- Sharp theorem on the maximum number of edges in demand graphs.
- Enhanced conditions ensuring terminal-pairability in bipartite graphs.

## Abstract

We investigate the terminal-pairibility problem in the case when the base graph is a complete bipartite graph, and the demand graph is also bipartite with the same color classes. We improve the lower bound on maximum value of $\Delta(D)$ which still guarantees that the demand graph $D$ is terminal-pairable in this setting. We also prove a sharp theorem on the maximum number of edges such a demand graph can have.