On Opsut’s conjecture for hypercompetition numbers of hypergraphs

The notion of the competition hypergraph was introduced as a variant of the notion of the competition graph by Sonntag and Teichert in 2004. They also introduced the notion of the hypercompetition number of a graph. In 1982, Opsut conjectured that for a locally cobipartite graph G, the competition number of G is less than or equal to 2 and the equality holds if and only if the vertex clique cover number of the neighborhood of v is exactly 2 for each vertex v of G. Despite the various attempts to settle the conjecture, it is still open. A hypergraph version of the Opsut’s conjecture can be stated as the assertion that for a hypergraph H, if the number of hyperedges containing v is at most 2 for each vertex v of H, then the hypercompetition number of H is less than or equal to 2 and the equality holds if and only if the number of hyperedges containing v is exactly 2 for each vertex v of H. In this paper, we show that this hypergraph version is true.