A relaxation of Novosibirsk 3-color conjecture

The famous Steinberg's conjecture states that planar graphs without cycles of lengths 4 and 5 are (0,0,0)-colorable. Recently, Cohen-Addad et al. [6] demonstrated that Steinberg's conjecture is false by constructing a counterexample. Let F denote the family of planar graphs without 3-cycles adjacent to cycles of lengths 3 and 5. Borodin et al. posed the Novosibirsk 3-color conjecture, which is the statement that every graph in F is (0,0,0)-colorable. It is easy to observe that the counterexample of Cohen-Addad et al. shows also that if G∈F, then G is not always (0,0,0)-colorable. Motivated by this observation, this paper proves that every member G∈F is (1,1,0)-colorable, which is a positive step.