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. 



A different version of list coloring
This problem can indeed be solved using online list coloring. The stronger result that you can use lists of size $\max(\deg(v\_i),\deg(v\_j))$ is proven as Theorem 3.3 in this [paper](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r77/pdf) of Uwe Schauz.


