A (3,1)? ( 3 , 1 ) ? -choosable theorem on toroidal graphs ?
An (L,d)? ( L , d ) ? -coloring c o l o r i n g is a mapping ? ? that assigns a color ?(v)?L(v) ? ( v ) ? L ( v ) to each vertex v?V(G) v ? V ( G ) such that at most d d neighbors of v v receive color ?(v) ? ( v ) . A graph G G is called (k,d)? ( k , d ) ? -choosable if it admits an (L,d)? ( L , d ) ? -coloring for every list assignment L L with |L(v)|?k | L ( v ) | ? k for all v?V(G) v ? V ( G ) . Let G G be a graph embeddable on the torus. In this paper, it is proved that G G is (3,1)? ( 3 , 1 ) ? -choosable if G G contains no 5- and 6-cycles.
toroidal graph cycle improper choosability