Every planar graph without 4-cycles and 5-cycles is (3,3)-colorable.

A graph is (d 1 , … , d k) -colorable if the vertex set can be partitioned into k sets V 1 , … , V k where the maximum degree of the graph induced by V i is at most d i for each i, where 1 ≤ i ≤ k . In this paper, we prove that every planar graph without 4-cycles and 5-cycles is (3,3)-colorable, which improves the result of Sittitrai and Nakprasit, who proved that every planar graph without 4-cycles and 5-cycles is (3, 5)-colorable (Sittitrai and Nakprasit in Discrete Math 341:2142–2150, 2018). [ABSTRACT FROM AUTHOR]