  The notion of a Galvin orientation of a line graph is introduced,
generalizing the idea used by Galvin in his landmark proof of the
list-edge-colouring conjecture for bipartite graphs. If L(G) has a proper
Galvin orientation with respect to k, then it immediately implies that G is
k-list-edge-colourable, but the converse is not true. The stronger property is
studied in graphs of the form `bipartite plus an edge', the Petersen graph,
cliques, and simple graphs without odd cycles of length 5 or longer.
