Improper coloring of toroidal graphs

A graph $G$ is called $(d_1,\dots,d_k)$-colorable if its vertices can be partitioned into $k$ sets $V_1,\dots,V_k$ such that $\Delta(\langle V_i\rangle_G)\leq d_i, i\in \{1,\dots, k\}$. If $d_1 = \dots = d_k = m$ we say that $G$ is $k$-colorable with defect $m$. A coloring with at least one $d_i, i\in \{1,\dots, k\}$, 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 $K_7$ and $T_{11}$ are $(0,0,0,1)$-colorable. Finally, we discuss the colorability of graphs embeddable on $N_1$ and show that they are $(0,0,0,2)$-colorable.