Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: If the vertex set of a graph with maximal degree Delta is partitioned into sets V-i of size 2 Delta, then there exists a coloring of the graph by 2 Delta colors, where each color class meets each V-i at precisely one vertex. We shall name it the strong 2 Delta-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR's). En route, we give a survey of some recent developments in the theory of ISR's.