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

Abstract

A graph is (d(1), . . . ,d(r))-colorable if its vertex set can be partitioned into r sets V-1, . . . ,V-r where the maximum degree of the graph induced by V-1, is at most d(i) for each i is an element of{1, . . . ,r}. Let g(g) denote the class of planar graphs with minimum cycle length at least g. We focus on graphs in g(5) since for any d(1) and d(2), Montassier and Ochem constructed graphs in g(4) that are not (d(1), d(2))-colorable. It is known that graphs in g(5) are (2, 6)-colorable and (4, 4)-colorable, but not all of them are (3, 1)-colorable. We prove that graphs in g(5) are (3, d(2))-colorable, leaving two interesting questions open: (1) are graphs in g(5) also (3, d(2))-colorable for some d(2) is an element of {2, 3, 4}? (2) are graphs in g(5) indeed (d(1), d(2))-colorable for all d(1) + d(2) >= 8 where d(2) >= d(1) >= 1?