The Erdős–Gallai theorem is a fundamental result in graph theory that pertains to the characterization of graphs with a given number of edges. Specifically, it provides a criterion for deciding whether a graph can exist with a specified number of edges and vertices, while also satisfying certain degree conditions.
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The Erdős-Gallai theorem characterizes which finite nonincreasing sequences of nonnegative integers form the degree sequence of a simple graph, using an even total degree and partial-sum inequalities.