On the list chromatic index of nearly bipartite multigraphs

Galvin ([7]) proved that every k-edge-colorable bipartite multigraph is kedge-choosable. Slivnik ([11]) gave a streamlined proof of Galvin's result. A multigraph G is said to be nearly bipartite if it contains a special vertex Vs such that G Vs is a bipartite multigraph. We use the technique in Slivnik's proof to obtain a list coloring analog of Vizing's theorem ([12]) for nearly bipartite multigraphs, and to obtain an extension (suggested by Woodall ([13])) of Galvin's result to multigraphs whose underlying simple graph is bipartite 'plus one edge'. We also prove that for any nearly bipartite multigraph G with special vertex Vs of degree at most six, if G is k-edge-colorable then G is k-edge-choosable.