Papers
This section presents a collection of AHP/ANP papers. Whether you are a student, academic and/or professional in the field of decision making, you can help us expand this collection by sharing your papers on decision making with the Analytic Hierarchy Process (AHP) and the Analytic Network Process (ANP).
Search papers by keyword:
Paper results for keyword: Complete graphThe minimum number of intersections in complete graphs
Thomas SaatyJournal: Proceedings of the National Academy of Sciences of the United States of AmericaZarankiewicz has developed results concerning the minimum number of edge intersections, when drawn in a plane, of the bipartite graph consisting of two sets of vertices with one edge joining each element of one set to each element of the other set. When each set has three vertices, we have one of...
Read MoreTwo theorems on the minimum number of intersections for complete graphs
Thomas SaatyJournal: Journal of Combinatorial TheoryAfter summarizing results of work on a conjecture regarding the minimum number of intersections of a complete graph drawn in the plane, the paper gives two theorems: one regarding a realization scheme for the conjectured quantity, and the other regarding a proof that essentially, the result would...
Read MoreRemarks on the four color problem; the Kempe catastrophe
Thomas SaatyJournal: Mathematics MagazineThe original error discovered by Heawood in Kempe's attempt to prove the four-color conjecture is often encountered by many of those who follow the inductive argument approach to the problem. Perhaps the nature of the difficulty is not well appreciated. An old Chinese proverb urges that to know t...
Read MoreOn polynomials and crossing numbers of complete graphs
Thomas SaatyJournal: Journal of Combinatorial TheoryA long-standing, unsolved problem is that of finding the minimum number of crossings of the edges in a complete graph when embedded on a surface of genus zero.
Read More
The minimum number of intersections in complete graphs
Thomas SaatyJournal: Proceedings of the National Academy of Sciences of the United States of AmericaZarankiewicz has developed results concerning the minimum number of edge intersections, when drawn in a plane, of the bipartite graph consisting of two sets of vertices with one edge joining each element of one set to each element of the other set. When each set has three vertices, we have one of...
Read MoreTwo theorems on the minimum number of intersections for complete graphs
Thomas SaatyJournal: Journal of Combinatorial TheoryAfter summarizing results of work on a conjecture regarding the minimum number of intersections of a complete graph drawn in the plane, the paper gives two theorems: one regarding a realization scheme for the conjectured quantity, and the other regarding a proof that essentially, the result would...
Read MoreRemarks on the four color problem; the Kempe catastrophe
Thomas SaatyJournal: Mathematics MagazineThe original error discovered by Heawood in Kempe's attempt to prove the four-color conjecture is often encountered by many of those who follow the inductive argument approach to the problem. Perhaps the nature of the difficulty is not well appreciated. An old Chinese proverb urges that to know t...
Read MoreOn polynomials and crossing numbers of complete graphs
Thomas SaatyJournal: Journal of Combinatorial TheoryA long-standing, unsolved problem is that of finding the minimum number of crossings of the edges in a complete graph when embedded on a surface of genus zero.
Read More