Localized Version of Hypergraph Erdos-Gallai Theorem

This paper focuses on extensions of the classic Erd{o}s-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge e of an n-vertex uniform hypergraph H is defined to a special function with respect to the number of edges of the longest Berge path containing e. We prove that the summation of the weighted function of all edges is at most n for an n-vertex uniform hypergraph H and characterize all extremal hypergraphs that attain the value, which strengthens and extends the hypergraph version of the classic Erd{o}s-Gallai Theorem.