Planar graphs with girth at least 55 are (3,5)(3,5)-colorable

A graph is (d1,…,dr)(d1,…,dr)-colorable if its vertex set can be partitioned into rr sets V1,…,VrV1,…,Vr where the maximum degree of the graph induced by ViVi is at most didi for each i∈{1,…,r}i∈{1,…,r}. Let GgGg denote the class of planar graphs with minimum cycle length at least gg. We focus on graphs in G5G5 since for any d1d1 and d2d2, Montassier and Ochem constructed graphs in G4G4 that are not (d1,d2)(d1,d2)-colorable. It is known that graphs in G5G5 are (2,6)(2,6)-colorable and (4,4)(4,4)-colorable, but not all of them are (3,1)(3,1)-colorable. We prove that graphs in G5G5 are (3,5)(3,5)-colorable, leaving two interesting questions open: (1) are graphs in G5G5 also (3,d2)(3,d2)-colorable for some d2∈{2,3,4}d2∈{2,3,4}? (2) are graphs in G5G5 indeed (d1,d2)(d1,d2)-colorable for all d1+d2≥8d1+d2≥8 where d2≥d1≥1d2≥d1≥1?