Regular colorings in regular graphs

An (r−1,1)-coloringof anr-regular graphGis an edge coloring (witharbitrarily many colors) such that each vertex is incident tor−1 edgesof one color and 1 edge of a different color. In this paper, we completelycharacterize all 4-regular pseudographs (graphs that may contain paralleledges and loops) which do not have a (3,1)-coloring. Also, for eachr≥6 we construct graphs that are not (r−1,1)-colorable and, more generally, arenot (r−t, t)-colorable for smallt.