Zarankiewiczʼs Conjecture is finite for each fixed m

Abstract

Zarankiewicz's Crossing Number Conjecture states that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals Z(m, n) := m/2 (m − 1)/2 n/2 (n − 1)/2 , for all positive integers m, n. This conjecture has only been verified for min{m, n} ≤ 6, for K 7,7 , K 7,8 , K 7,9 , and K 7,10 , and for K 8,8 , K 8,9 , and K 8,10 . We determine, for each positive integer m, an integer N 0 = N 0 (m) with the following property: if cr(Km,n) = Z(m, n) for all n ≤ N 0 , then cr(Km,n) = Z(m, n) for every n. This yields, for each fixed integer m, a finite algorithm that either proves that cr(Km,n) = Z(m, n) for every n, or else finds a counterexample.

Sketchy Tweets: Ten Minute Conjectures in Graph Theory

“…Hence, the smallest unsolved cases are for K 7,11 and K 9,9 . As b A recent result in [7] states that if for a fixed m, the conjecture holds for all values n smaller than some constant depending on m, then the conjecture holds for all n. Hence, for each m there is an algorithm that verifies the conjecture for all n or gives a counterexample.…”

Exact Crossing Number Parameterized by Vertex Cover

“…The same idea, only for the special case of the complete bipartite graph G = K k,n , has already been used by Christian, Richter and Salazar [8] (their research goal, though, was different). Our paper can be considered a generalization of (a part of) [8]. To achieve our goal, we separate two kinds of crossings and rigorously describe a topological clustering of a drawing of G.…”

“…Let I be an index set defined as I := {(i, j) | 1 ≤ i ≤ l, 1 ≤ j ≤ g(i)}. Similarly as in [8], we define the following crossing vector p = (p α | α ∈ I) and the crossing matrix Q = (q α,β | α, β ∈ I), such that the intended use of p is to count the crossings between the edges of G X and the edges incident to each topological cluster corresponding to a vertex of V (C X ) \ X, and the intended use of Q is to count the cluster crossings of each one of the topological clusters (the diagonal entries) and the non-cluster crossings between pairs of the clusters (the other entries):…”

On the Crossing Number of Kn without Computer Assistance

“…Place a new vertex v near v in D, join v to v, and, for each edge vw of K n , draw v w near D [vw]. (2) There are two types of new crossings created: there are crossings of edges v w with edges of K n − v; and there are crossings of edges v w with edges vx. The number of crossings of the first type is equal to the number of crossings of vw with edges of K n − v. The second type of crossings are the duplication costs.…”

“…The rotations of the edges around v and v are the same (except possibly for where the edge vv occurs). The following is easy to see and is done in [2] and [8]. Let If D [uv] is crossed only once and n ≥ 6, this leaves at least two crossings of these edges for D [uv].…”