Consider a graph formed by $ k $ k Order cliques sharing at most one point. Now, to color the graph, we first color any clique.
Now, if a top of the color clique is shared by $ m $ clicks, then, we could put to the maximum $ k- (m-1) $ vertices to develop the color class containing this vertex, which are chosen among the cliques that do not share this vertex. This corresponds to the number of reduced vertices of $ k ^ 2 $ the vertices, which is the degenerate case of all disjointed cliques. By doing so for all the vertices of the colored clique, I think we can cover all the vertices of the graph by developing the color classes. So, the graph as a whole would $ k- $ colorable.
Is the statement above correct? Or are there counterexamples? Any light on it. Thank you in advance.