Upper bound on chromatic number for some graphs 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.
 A: No, not every graph in your class is planar.  Let $G$ be the graph obtained from $K_{3,3}$ by adding a new vertex that is adjacent to all vertices of $K_{3,3}$.  Since, $K_{3,3}$ is triangle free, $G$ does not contain $K_4$.  Moreover, it is clear that every edge of $G$ is contained in a triangle.  
Your class of graphs does not have bounded chromatic number either.  For example, the Mycielski Graphs are a sequence of triangle-free graphs that have arbitrarily large chromatic number.  By adding an apex vertex to the Mycielski Graphs (or performing Pat Devlin's construction) you get a sequence of graphs that have unbounded chromatic number and satisfy your conditions.  
A: 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.) 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. 
A: (First, welcome to graph theory)
No, there's likely not much you can say by adding your restriction (that every edge is on a triangle).
Consider for example the following construction.
Let $G$ be any graph whatsoever (and make $G$ something ugly where you feel like you have no control over the chromatic number).
Then for each edge, create a new vertex and attach it to both ends of that edge (thereby putting every edge on a triangle).
This doesn't change the chromatic number [provided $G$ wasn't bipartite] or the clique number [provided $G$ had triangles], and it changes the max degree by doubling it.  So anything you had controlling the chromatic number shouldn't involve parameters unaffected by this operation.
