I am a beginner in graph theory and I am interested in finding an upper bound for the chromatic number of the following class of graphs:

- If two vertices $a$ and $b$ are adjacent in $G$, then there exist vertex $c$ such that $abc$ is a triangle graph. In other words, there exist vertex $c$ such that $c$, $a$ and $b$ are adjacent.
- Every graph in this class has no complete sub-graph except $K_3$.

For example chromatic number of triangle is $\chi(G)=3$. In general, an upper bound for the chromatic number of an arbitrary graph $G$ is $\Delta(G)+1$. But this bound is not necessarily optimal for the above problem.

Is there any paper about this problem?

**Update:** Is this class of graphs planar? If not, under what additional conditions is this class of graphs planar?

Thanks.