Packing chromatic number versus chromatic and clique number

Abstract

The packing chromatic number $$\chi _{\rho }(G)$$ of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets $$V_i$$ , $$i\in [k]$$ , where each $$V_i$$ 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 $$\omega (G) = a$$ , $$\chi (G) = b$$ , and $$\chi _{\rho }(G) = c$$ . If so, we say that (a, b, c) is realizable. It is proved that $$b=c\ge 3$$ implies $$a=b$$ , and that triples $$(2,k,k+1)$$ and $$(2,k,k+2)$$ are not realizable as soon as $$k\ge 4$$ . 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 on $$\chi _{\rho }(G)$$ in terms of $$\Delta (G)$$ and $$\alpha (G)$$ is also proved.