On Finding Lagrangians of \(3\)-uniform Hypergraphs

Abstract

It is known that determining the Lagrangian of a general \(r\)-uniform hypergraph is useful in practice and is non-trivial when \(r \geq 3\). 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 \(m = \binom{k}{3} – a\), 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.