On the cone of weighted graphs generated by triangles

Motivated by problems involving triangle-decompositions of graphs, we examine the facet structure of the cone $\tau_n$ of weighted graphs on $n$ vertices generated by triangles. Our results include enumeration of facets for small $n$, a construction producing facets of $\tau_{n+1}$ from facets of $\tau_n$, and an arithmetic condition on entries of the normal vectors. We also point out that a copy of $\tau_n$ essentially appears via the perimeter inequalities at one vertex of the metric polytope.