Edge-coloring almost bipartite multigraphs

Bipartite multigraphs have chromatic index equal to the largest degree d. We consider multigraphs obtained by inserting k vertices in edges of a connected bipartite multigraph, and show that the chromatic index may increase to at most d+1. We further show that (1) the chromatic index always increases if k=1 when the original bipartite multigraph is regular, but does not increase for k=1 if the bipartite multigraph is not regular; (2) the chromatic index does not increase if k=2 (regular or not); (3) the chromatic index may increase if k=3 when the original bipartite multigraph is regular, but does not increase for k=3 if the bipartite multigraph is not regular; (4) the chromatic index does not increase if k=4 (regular or not); (5) the chromatic index increases for infinitely many 3-regular bipartite graphs and each k=3 odd, and (6) the chromatic index does not increase for infinitely many 3-regular bipartite graphs and each k=3 (even or odd). We finally study the list chromatic index of such multigraphs and show that it does not increase past d+1 either, in fact in some sense it stays almost d. All the results presented here render polynomial-time algorithms, since the proofs are based on bipartite graph matching and stable marriage.