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

Hiroaki Mori

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 $k=\lceil\frac{n}{2}\rceil$.

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

Abstract

For a graph , let be the largest distortion necessary to embed any shortest-path metric on into , and for any natural number , denote as the complete bipartite graph. In this note, we caculate the value of , more precisely we prove where .
Paper Structure (3 sections, 5 theorems, 30 equations, 1 figure, 2 algorithms)

This paper contains 3 sections, 5 theorems, 30 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lower Bounds
  3. Upper Bounds

Key Result

Theorem 1

The maximum ratio between the sparsest cut and the maximum concurrent flow ranges over all instances on $G$ matches $c_1(G)$.

Figures (1)

  • Figure 1: Graph $K_{2,2k}^{\ell}$ where $k=2, \ell=3$

Theorems & Definitions (5)

  • Theorem 1: Gutpa et al. Gupta2004
  • Theorem 2
  • Theorem 3: Hypermetric inequalityDeza
  • Proposition 4
  • Proposition 5