A kind of online edge-coloring problems on bipartite graphs is considered. The input is a graph (typically with no edges) and a sequence of operations (edge addition and edge deletion) under the restriction that at any time the graph is bipartite and degree-bounded by k, where k is a prescribed integer. At the time of edge addition, the added edge can be irrevocably assigned a color or be left uncolored. No other coloring or color alteration is allowed. The problem is to assign colors as many times as possible using k colors. Two algorithms are presented: one with competitiveness coefficient 1/4 against oblivious adversaries, and one with competitiveness coefficient between 1/4 and 1/2 with the cost of requiring more random bits than the former algorithm, also against oblivious adversaries.