Planar graphs without cycles of length 4 or 5 are -colorable

We study Steinberg’s Conjecture. A graph is ( c 1 , c 2 , … , c k ) -colorable if the vertex set can be partitioned into k sets V 1 , V 2 , … , V k such that for every i with 1 ≤ i ≤ k the subgraph G [ V i ] has maximum degree at most c i . We show that every planar graph without 4- or 5-cycles is ( 3 , 0 , 0 ) -colorable. This is a relaxation of Steinberg’s Conjecture that every planar graph without 4- or 5-cycles is properly 3-colorable (i.e.,  ( 0 , 0 , 0 ) -colorable).