Packing chromatic number, (1,1,2,2)-colorings, and characterizing the Petersen graph

The packing chromatic number χ ρ ( G ) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Π 1 , … , Π k , where Π i , i ∈ [ k ] , is an i -packing. The following conjecture is posed and studied: if G is a subcubic graph, then χ ρ ( S ( G ) ) ≤ 5 , where S ( G ) is the subdivision of G . The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if G is a generalized prism of a cycle, then G is (1, 1, 2, 2)-colorable if and only if G is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two n -cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered.