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:



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> 1. 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.
> 2. Every graph in this class has no complete sub-graph except $K\_3$.
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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.



Upper bound on chromatic number for some graphs
For any $H$ (in particular for $K\_4$) there exist a constant $c(H)$ such that the chromatic number of any $H$-free graph $G$ does not exceed $c(H)\frac{d\log\log d}{\log d}$, $d=\Delta(G)$ (provided that $d>0$). It is [(conjecture 3.1.)](https://www.tau.ac.il/~nogaa/PDFS/logf4.pdf) conjectured (or already proved? oir disproved? I do not know) that double logarithm may be removed. This is proved for $H=K\_3$ by Johansson. The proof is quite difficult. 


