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. [ABSTRACT FROM AUTHOR]