toroidal graph; defective choosability; chord
A graph $G$ is called $(k,d)^*$-choosable if for every list assignment $L$ satisfying $|L(v)|= k$ for all $v\in V(G)$, there is an $L$-coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored with the same color as itself. In this paper, it is proved that every toroidal graph without chordal 7-cycles and adjacent 4-cycles is $(4,1)^*$-choosable.
 Eaton N., Hull T.: Defective list colorings of planar graphs
. Bull. Inst. Comb. Appl. 25 (1999), 79–87. MR 1668108
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