On a conjecture of Erd?s, Rubin and Taylor

A graph G with vertex set V and edge set E is called (a; b)-choosable if for any assignment of lists L(v) of colors to the vertices v of G with jL(v)j = a it is possible to choose subsets C(v) L(v), jC(v)j = b for every v 2 V such that C(u) \ C(v) = ; for all uv 2 E. In 1979, Erd} os, Rubin and Taylor raised the conjecture that every (a; b)-choosable graph G is also (am; bm)-choosable for all m > 1. We investigate the case b = 1, presenting some classes of (a; 1)-choosable graphs for which (am; m)-choosability can be proved for all m.