Finding a class of graphs for which vertex coloring is NP-hard We know that for general graphs vertex coloring problem/ minimum clique cover problem are NP-hard and for perfect graphs, they can be solved in polynomial time. There are classes of graphs which are perfect, e.g., bipartite graphs, chordal graphs, linear graphs, etc.

My question is that:

Are there any families of graphs that we know for them vertex coloring problem/ minimum clique cover problem are NP-hard? I appreciate if someone can give the name of some of them or any reference for them (in case they exist).

I have a problem and I want to show that it is hard and I want to find such a family of graphs and use them as an instance for my problem.
This PDF defines a graph for which solving a 3-SAT formula is equivalent to determining if the graph is 3-colorable. Finding the vertex coloring of these graphs would solve the 3-SAT problem, so the vertex coloring problem on these graphs is at least as hard as an NP-complete problem. Therefore, solving the vertex coloring problem on these graphs is NP-hard.