Find an original example of a graph whose chromatic number does not equal its clique number, yet whose clique partition number equals its independence number.
Chromatic number: $\chi(G)$ is the minimum colors needed to color a graph
Clique number: $\omega (G)$ is the maximum number of vertices of a complete subgraph of the graph
Independence number: $\beta (G)$ is the largest set of independent vertices
Clique partition: least number of cliques that partition the vertex set
*I know what each component means and how to find it but I can't find an example that fits all of the pieces of criteria. Right now I am just trying random graphs- is there a systematic ways to do this?
