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 GG, the competition number of GG is less than or equal to 22 and the equality holds if and only if the vertex clique cover number of the neighborhood of vv is exactly 22 for each vertex vv of GG. 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 HH, if the number of hyperedges containing vv is at most 22 for each vertex vv of HH, then the hypercompetition number of HH is less than or equal to 22 and the equality holds if and only if the number of hyperedges containing vv is exactly 22 for each vertex vv of HH. In this paper, we show that this hypergraph version is true.