List coloring conjecture

The List Coloring Conjecture states that every multigraph has list edge chromatic number equal to its edge chromatic number.

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Edge coloring: An assignment of colors to a graph’s edges in which edges sharing a vertex receive different colors. Its minimum color count is the ordinary parameter the conjecture compares.
Maximum degree: The largest number of edges incident to any vertex in a graph. It gives a universal lower bound on both edge-coloring parameters.
List coloring: A graph coloring problem in which each vertex has an individual list of permitted colors. The edge version adapts this general coloring problem to incident edges.
Edge-chromatic class: A classification dividing multigraphs according to whether their edge chromatic number equals maximum degree or exceeds it. The conjecture must hold in both classes, though their coloring behavior differs.
List edge coloring: An edge coloring in which each edge must receive a color from its own prescribed list. Its minimum list size is the other parameter in the conjecture.
Vizing's theorem: A theorem stating that a simple graph’s edge chromatic number is either its maximum degree or one greater. It sharply bounds the ordinary parameter for simple graphs.
Paul Erdős: A Hungarian mathematician whose work shaped graph theory, combinatorics, and probability. Erdős introduced the list-coloring conjecture with collaborators.
Overfull graph: A graph whose edge count exceeds the maximum possible for a proper coloring with its maximum degree colors. Overfull subgraphs can force edge chromatic number above maximum degree.
Edge chromatic number: The minimum number of colors needed for a proper edge coloring of a graph. The conjecture claims lists never require more colors than this number.
Shannon's theorem: A theorem bounding the edge chromatic number of a multigraph by three halves of its maximum degree. Its multigraph bound offers a benchmark for the conjecture’s setting.