Consider a non-regular bipartite graph $G$ . We consider list edge coloring the edges of the graph by giving lists of cardinality $max(deg(v_i),deg(v_j))+2$ for each edge $e=v_iv_j$ where $deg(v_i)$ denotes the degree of the vertex $v_i$.

Then, is the graph $G$ properly colorable. If so, would the maximum cardinality of the lists be $\Delta+2$ where $\Delta$ be the maximum degree of $G$? Is this version of list coloring of $G$ similar to online list coloring? I also see it is exactly somewhat similar to the concept of $f-$ chhosabality of the graphs, where $f$ is the choice function for edge. Thanks beforehand.