On Opsuts 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 G , the competition number of G G is less than or equal to 2 2 and the equality holds if and only if the vertex clique cover number of the neighborhood of v v is exactly 2 2 for each vertex v v of G G . Despite the various attempts to settle the conjecture, it is still open. A hypergraph version of the Opsuts conjecture can be stated as the assertion that for a hypergraph H H , if the number of hyperedges containing v v is at most 2 2 for each vertex v v of H H , then the hypercompetition number of H H is less than or equal to 2 2 and the equality holds if and only if the number of hyperedges containing v v is exactly 2 2 for each vertex v v of H H . In this paper, we show that this hypergraph version is true.
opsuts conjecture hypercompetition number competition number competition hypergraph competition graph