Abstract
AbstractWe call a graph a ‐threshold graph if there are distinct real numbers and a mapping such that for any two vertices , we have that if and only if there are odd numbers such that . The least integer such that is a ‐threshold graph is called a threshold number of , and denoted by . The well‐known family of threshold graphs is a set of graphs with . Jamison and Sprague introduced the concept of ‐threshold graph, and proved that exists for every graph . They further obtained a number of interesting results on . In addition, they also proposed several unsolved problems and conjectures, including the following two. Problem: Determine the exact threshold numbers of the complete multipartite graphs. Conjecture: For all even , there is a graph with and . This is equivalent to that for all odd , there is a graph with and , where is the complement of .In this short paper, we give a partial solution of the problem and confirm the conjecture.
Published Version
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