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.
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).