On Cliques and Lagrangians of 3-uniform Hypergraphs

TL;DR

This paper investigates the relationship between the Lagrangian and maximum clique size in 3-uniform hypergraphs, providing evidence for a conjecture and proposing an algorithm to verify it, extending classical graph results.

Contribution

It extends the Motzkin-Straus connection to 3-uniform hypergraphs and offers a combinatorial algorithm to test a related conjecture.

Findings

Evidence supporting the conjecture for certain hypergraph ranges

A new combinatorial algorithm for conjecture verification

Insights into the Lagrangian-clique relationship in hypergraphs

Abstract

There is a remarkable connection between the maximum clique number and the Lagrangian of a graph given by T. S. Motzkin and E.G. Straus in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we explore evidences that the Lagrangian of a 3-uniform hypergraph is related to the order of its maximum cliques when the number of edges of the hypergraph is in certain range. In particular, we present some results about a conjecture introduced by Y. Peng and C. Zhao (2012) and describe a combinatorial algorithm that can be used to check the validity of the conjecture.