Planar graphs with girth at least 5 are (3, 5)-colorable

A graph is (d1, . . . , dr )-colorable if its vertex set can be partitioned into r sets V1, . . . , Vr where themaximum degree of the graph induced by Vi is at most di for each i ∈ {1, . . . , r}. Let Gg denote the class of planar graphs with minimum cycle length at least g . We focus on graphs in G5 since for any d1 and d2, Montassier and Ochem constructed graphs in G4 that are not (d1, d2)-colorable. It is known that graphs inG5 are (2, 6)-colorable and (4, 4)colorable, but not all of them are (3, 1)-colorable. We prove that graphs in G5 are (3, 5)colorable, leaving two interesting questions open: (1) are graphs inG5 also (3, d2)-colorable for some d2 ∈ {2, 3, 4}? (2) are graphs in G5 indeed (d1, d2)-colorable for all d1 + d2 ≥ 8 where d2 ≥ d1 ≥ 1? © 2014 Elsevier B.V. All rights reserved.