We prove a necessary and sufficient condition for the existence of edge list multicoloring of trees. The result extends the HalmosVaughan generalization of Hall's theorem on the existence of distinct representatives of sets. ß
Let n(G) denote the number of vertices of a graph G and let (G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined bywhere the maximum is taken over all induced subgraphs H of G. It is obvious that every graph G satisfies (G) (G) (G) where and denote the clique number and the chromatic number of G, respectively. We show that the interval [ (G), (G)] can be arbitrary large by estimating the Hall ratio of the Mycielski graphs.
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