List Edge and List Total Colourings of Multigraphs

This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B 63 (1995), 153 158), who proved that the list edge chromatic number /$list(G) of a bipartite multigraph G equals its edge chromatic number /$(G). It is now proved here that if every edge e=uw of a bipartite multigraph G is assigned a list of at least max[d(u), d(w)] colours, then G can be edge-coloured with each edge receiving a colour from its list. If every edge e=uw in an arbitrary multigraph G is assigned a list of at least max[d(u), d(w)]+w 2min[d(u), d(w)]x colours, then the same holds; in particular, if G has maximum degree 2=2(G) then /$list(G) w 2 2x . Sufficient conditions are given in terms of the maximum degree and maximum average degree of G in order that /$list(G)=2 and /"list(G)=2+1. Consequences are deduced for planar graphs in terms of their maximum degree and girth, and it is also proved that if G is a simple planar graph and 2 12 then /$list(G)=2 and /"list(G)=2+1. 1997 Academic Press