The exact value of c1(K2,n)

For a graph G , let c 1 ( G ) be the largest distortion necessary to embed any shortest-path metric on G into ℓ 1 , and for any natural number n , m ∈ N , denote K n , m as the complete bipartite graph. In this note, we calculate the value of c 1 ( K 2 , n ) , more precisely we prove c 1 ( K 2 , n ) = 3 k − 2 2 k − 1 where k = ⌈ n 2 ⌉ .