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.
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