Graphs whose maximal clique intersection graph has bounded chromatic number In general,
The chromatic number of the clique intersection graph a graph can again be arbitrarily high. 
For example, take $n$ maximal cliques of size $n$ such that every pair of maximal cliques has a vertex in common.

So, for which non-trivial graphs the clique intersection graphs has bounded chromatic number even if the graph can have arbitrarily high chromatic number?


Any reference to surveys in this area is welcome. 
I welcome any reference relating the graphs with bounded degree and to the chromatic number of its clique intersection graph.
Let the independent neighborhood of a vertex $v$ in a graph is the maximum independent set with every vertex having a common neighbor as $v$. 
Let for a graph family $F$, we have a constant $C(F)$, such that for any graph in the family $F$, the size of independent neighborhood of a vertex $v$ in the graph is bounded by $C(F)$. Are there any studies which relate the chromatic number of the clique intersection graph family of $F$ with the above notion.


Edit: 

In the clique intersection graph $C(G)$ of a graph $G$, the nodes are maximal cliques of $G$ and there is an edge whenever they share a vertex.

Note: This is different from the definition of clique-separator graph for chordal graphs (Which is a tree).
For (2), any maximal clique $C$ of $G$ has size at most $\Delta+1$ and each vertex $v \in C$ participates in at most $2^{|N(v)|} \leq 2^\Delta$ maximal cliques. So the degree of $C$ in the clique intersection graph is at most $|C| \cdot 2^{|N(v)|} \leq (\Delta + 1) \cdot 2^\Delta$. Thus the clique intersection graph can be $(\Delta + 1) \cdot 2^\Delta+1$ colored. 

The bound can be tightened to  $(\Delta + 1) \cdot 3^{\Delta/3}+1$ by using the Moon Moser bound for the number of maximal cliques.