The following graph property has come up naturally in some work I've been doing, and it seems like something that may have already been studied.

Namely, let $G$ be a graph with no loops or double edges, such that given any two edges $e, e'$, there exist triangles $T_1, ..., T_n$ in $G$ such that $e\in T_1, e'\in T_n$, and $T_i$ and $T_{i+1}$ have an edge in common for all $1 \leq i < n$. Does this type of graph have a name?

Equivalently, let $T(G)$ be the graph whose vertices are the edges of $G$, with an edge $e\sim e'$ iff $e$ and $e'$ are both contained in some triangle of $G$. Then $G$ is of the type above iff $T(G)$ is connected.

In particular, I am interested in research on the chromatic numbers of such graphs.

EDIT: I figured I'd expand a bit on motivation with respect to Gjergji Zaimi's answer. Namely, I'm working on machinery which is able to reduce questions about coloring a graph $G$ to questions about coloring minor subgraphs of the type I've described. The point is that these graphs are "rigid" in the sense that contracting the edges of any proper subgraph gives rise to loops or double edges. So essentially I'm after sharp results that bound from above the chromatic numbers of such graphs in terms of properties unrelated to edge contraction--in particular bounds in terms of *local* properties (e.g. the existence of induced subgraphs of certain types) would be ideal.