On relations between transportation cost spaces and ${\mathrm{\ell }}_1$

The present paper deals with some structural properties of transportation cost spaces, also known as Arens-Eells spaces, Lipschitz-free spaces and Wasserstein spaces. The main results of this work are: (1) A necessary and sufficient condition on an infinite metric space $M$, under which the transportation cost space on $M$ contains an isometric copy of ${\mathrm{\ell }}_1$. The obtained condition is applied to answer the open questions asked by C'uth and Johanis (2017) concerning several specific metric spaces. (2) The description of the transportation cost space of a weighted finite graph $G$ as the quotient ${\mathrm{\ell }}_1(E(G))\mathrm{/}Z(G)$, where $E(G)$ is the edge set and $Z(G)$ is the cycle space of $G$. This is a generalization of the previously known result to the case of any finite metric space.