On Finding Lagrangians of 3-uniform Hypergraphs.

It is known that determining the Lagrangian of a general r-uniform hypergraph is useful in practice and is non-trivial when r >= 3. In this paper, we explore the Lagrangians of 3-uniform hypergraphs with edge sets having restricted structures. In particular, we establish a number of optimization problems for finding the largest Lagrangian of 3-uniform hypergraphs with the number of edges m = ((k)(3)) - a where a=3 or 4. We also verify that the largest Lagrangian has the colex ordering structure for 3-uniform hypergraphs when the number of edges is small.