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Does anyone know what are called (if there is any nomenclature for this class of graphs in the literature) the connected graphs such that each of their edge belongs to some triangle? For example, every complete graph with more than two vertices satisfies this property. Is there any reference to the subject?

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    $\begingroup$ triangle-covered $\endgroup$
    – Brendan McKay
    Commented Feb 17, 2020 at 22:08

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I've found this in the literature: "The k-truss of G is the largest subgraph of G in which every edge is contained in at least (k-2) triangles within the subgraph.". See:

So your graph needs to be a connected 3-truss.

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