The (3,3)-colorability of planar graphs without 4-cycles and 5-cycles

A graph G is called (d1, ... , dr)-colorable if its vertex set can be partitioned into r sets V1, ..., Vr such that the maximum degree of the induced subgraph G[Vi] of G is at most di for i is an element of {1, ... , r}. Steinberg conjectured that every planar graph without 4/5-cycles is (0, 0, 0)-colorable. Unfortunately, the conjecture does not hold and it has been proved that every planar graph without 4/5-cycles is (1, 1, 0)-colorable. When only two colors are allowed to use, it is known that some planar graphs without 4/5-cycles are not (1, k)-colorable for any k >= 0 and every planar graph without 4/5-cycles is (3, 4)-colorable or (2, 6)-colorable. In this paper, we reduce the gap for 2-coloring by proving that every planar graph without 4/5-cycles is (3, 3)-colorable. (c) 2022 Elsevier B.V. All rights reserved.