The exact value of $c_1(K_{2,n})$

TL;DR

This paper precisely calculates the largest distortion needed to embed the shortest-path metric of the complete bipartite graph K_{2,n} into  space, revealing a specific formula based on n.

Contribution

It provides an exact value for the distortion of embedding K_{2,n} into  space, which was previously unknown.

Findings

c_1(K_{2,n})=rac{3k-2}{2k-1} where k=rac{n}{2}

The result offers a precise metric distortion value for this class of bipartite graphs.

The formula depends explicitly on n, enabling exact calculations for any n.

Abstract

For a graph $G$, let $c_1(G)$ be the largest distortion necessary to embed any shortest-path metric on $G$ into $\ell_1$, and for any natural number $n,m\in\mathbb{N}$, denote $K_{n,m}$ as the complete bipartite graph. In this note, we caculate the value of $c_1(K_{2,n})$, more precisely we prove $c_1(K_{2,n})=\frac{3k-2}{2k-1}$ where $k=\lceil\frac{n}{2}\rceil$.