A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set A subset of N is less than B subset of N in the colex ordering if max(A Delta B)is an element of B. In 1980' s, Frankl and Furedi conjectured that the r-uniform graph with m edges consisting of the first m sets of N-(r) in the colex ordering has the largest Lagrangian among all runiform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r = 2. This conjecture seems to be challenging even for r = 3. For a hypergraph H = (V, E), the set T (H) = {|e| : e is an element of E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a nonuniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let Cm, T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H) <= L(C-m,C-T) hold? If T = {r}, then this question is the question by Frankl and Furedi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1, 2}-hypergraphs. Regarding the first question, we give a result for {1, r(1), r(2),..., r(l)}-hypergraph. We also show the connection between the generalized Lagrangian of {1, r(1), r(2),..., r(l)}-hypergraphs and {r(1), r(2),..., r(l)}-hypergraphs concerning the second question.