Each edge has a list of allowed colors; a list edge coloring chooses an allowed color on each edge, giving different colors to incident edges. The list-chromatic index is the uniform list-size threshold guaranteeing success.
The list-chromatic index is the least integer such that every assignment of at least allowed colors to each edge admits a list edge coloring. Uniform identical lists show it is at least the chromatic index. The Galvin theorem for bipartite list edge coloring gives equality for bipartite graphs.
A bipartite graph has equal list-chromatic index, chromatic index and maximum degree of a graph . Start with a proper -edge coloring. Orient its line graph toward smaller colors at one part and toward larger colors at the other. An edge of color has at most outgoing neighbours. Every induced orientation has a digraph kernel obtained from a stable matching in a bipartite graph with those preferences. Apply the kernel list-coloring lemma.
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List edge-coloring is a variation of the standard edge-coloring problem in graph theory, where each edge of a graph has a list of allowable colors from which it can be colored. The objective in list edge-coloring is to assign colors to the edges of the graph in such a way that: 1. No two adjacent edges (i.e., edges that share a common vertex) have the same color. 2. Each edge is colored using a color from its own list of allowable colors.