Packing chromatic number versus chromatic and clique number

The packing chromatic number of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets , , where each is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that , , and . If so, we say that (a, b, c) is realizable. It is proved that implies , and that triples and are not realizable as soon as . Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound chi(rho)(G) on in terms of Delta(G) and alpha(G) is also proved.