A (3,1)∗(3,1)∗-choosable theorem on toroidal graphs

An (L,d)∗(L,d)∗-coloringcoloring is a mapping ϕϕ that assigns a color ϕ(v)∈L(v)ϕ(v)∈L(v) to each vertex v∈V(G)v∈V(G) such that at most dd neighbors of vv receive color ϕ(v)ϕ(v). A graph GG is called (k,d)∗(k,d)∗-choosable if it admits an (L,d)∗(L,d)∗-coloring for every list assignment LL with |L(v)|≥k|L(v)|≥k for all v∈V(G)v∈V(G). Let GG be a graph embeddable on the torus. In this paper, it is proved that GG is (3,1)∗(3,1)∗-choosable if GG contains no 5- and 6-cycles.