A graph G is called (d₁, , dₖ) -colorable if its vertices can be partitioned into k sets V₁, , Vₖ such that Δ (VᵢG) dᵢ, i \1, , k\. If d₁ = = dₖ = m we say that G is k-colorable with defect m. A coloring with at least one dᵢ, i \1, , k\, greater than 0 is called an improper coloring. It is known that toroidal graphs are properly 7-colorable, therefore they are 7-colorable with defect 0. It was also proved that toroidal graphs are 5-colorable with defect 1 and 3-colorable with defect 2. The question whether they are 4-colorable with defect 1 remains open. In this paper we focus on improper coloring of toroidal graphs with values of defects being not all equal. We prove that these graphs are (0, 0, 0, 0, 0, 1^*) -colorable, (0, 0, 0, 0, 2) -colorable and (0, 0, 0, 1^*, 1^*) -colorable (a star means that there is an improper coloring in which subgraph induced by the corresponding color class contains at most one edge). Choi and Esperet in Improper coloring of graphs on surfaces, J. Graph Theory 91 (1) \, (2019), 16-34 proved that every graph of Euler genus eg > 0 is (0, 0, 0, 9eg - 4) -colorable. From this result it follows that toroidal graphs are (0, 0, 0, 14) -colorable. We decreased the value 14 and proved that toroidal graphs are (0, 0, 0, 4) -colorable. We also show that all 6-regular toroidal graphs except K₇ and T₁₁ are (0, 0, 0, 1) -colorable. Finally, we discuss the colorability of graphs embeddable on N₁ and show that they are (0, 0, 0, 2) -colorable.
Kolačkovská et al. (Fri,) studied this question.