# Terminal-Pairability in Complete Bipartite Graphs with Non-Bipartite Demands

## TL;DR

This paper studies the edge-disjoint paths problem in complete bipartite graphs with non-bipartite demand graphs, providing new bounds and extremal results on demand graph realizability.

## Contribution

It improves bounds on the maximum degree for guaranteed demand graph realization and determines the maximum number of edges ensuring realizability in complete bipartite graphs.

## Key findings

- Improved lower bounds on demand graph degree for realizability.
- Determined maximum edges guaranteeing demand graph realization.
- Enhanced understanding of the extremal conditions for terminal-pairability.

## Abstract

We investigate the terminal-pairability problem in the case when the base graph is a complete bipartite graph, and the demand graph is a (not necessarily bipartite) multigraph on the same vertex set. In computer science, this problem is known as the edge-disjoint paths problem. We improve the lower bound on the maximum value of $\Delta(D)$ which still guarantees that the demand graph $D$ has a realization in $K_{n,n}$. We also solve the extremal problem on the number of edges, i.e., we determine the maximum number of edges which guarantees that a demand graph is realizable in $K_{n,n}$.