On the crossing number of K m,n


 Randolf Rich
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1 On the crossing number of K m,n Nagi H. Nahas Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known on the crossing number of the complete bipartite graph is : cr(k m,n ) (1/5)(m)(m 1) n/ (n 1)/ In this paper we prove that: cr(k m,n ) (1/5)m(m 1) n/ (n 1)/ m n for sufficiently large m and n. 1 Introduction Determining the crossing number of the complete bipartite graph is one of the oldest crossing number open problems. It was first posed by Turan and known as Turan s brick factory problem. In 1954, Zarankiewicz conjectured that it is equal to Z(m, n) = n/ (n 1)/ m/ (m 1)/ He even gave a proof and a drawing that matches the lower bound, but the proof was shown to be flawed by Richard Guy [1]. Then in 1970 Kleitman proved that Zarankiewicz conjecture holds for Min(m, n) 6 []. In 199 Woodall proved it for m 8,n 10 []. Previously the best known lower bound in the general case was the one proved by Kleitman [] : cr(k m,n ) (1/5)(m)(m 1) n/ (n 1)/. Richter and Thomassen discussed the relation between the crossing numbers of the complete and the complete bipartite graphs [4]. the electronic journal of combinatorics 10 (00), #N8 1
2 A new bound We will start by giving definitions that will be used throughout the paper. They are taken from Woodall[] and Kleitman []. Definition 1 Two edges e 1 and e are said to have a crossing in a drawing D of K m,n if e 1 can be closed by a curve disjoint from e connecting the two endpoints of e 1 and such that there are points of e both inside and outside the closed curve. Definition The crossing number cr G of a graph G is the smallest crossing number of any drawing of G in the plane, where the crossing number cr D of a drawing D is the number of nonadjacent edges that have a crossing in the drawing. Definition A good drawing a graph G is a drawing where the edges are nonselfintersecting where each two edges have at most one point in common, which is either a common end vertex or a crossing. Clearly a drawing with minimum crossing number must be a good drawing. Let A be one partite and B the other partite. The elements of A are a 1,a,a,...,a m, and the elements of B are b 1,b,...,b n. In a drawing D, wedenotebycr D (a i,a k )the number of crossings of arcs, one terminating at a i, the other at a k and by cr D (a i )the number of crossings on arcs which terminate at a i : n cr D (a i )= cr D (a i,a k ) k=1 The crossing number of the drawing D is therefore: n n cr D = cr D (a i,a k ) i=1 k=i+1 Let us define : Z(m) = m/ (m 1)/. Let Sn be the set of the (n 1)! different cyclic orderings of a set V n of n elements. (The significance of cyclic ordering is that 014 is considered as being the same as 401 or 140). If z 1 and z belong to Sn then the distance d(z 1,z ) is the minimum number of transpositions between adjacent elements in the cyclic ordering necessary to turn z 1 into z. If a belongs to Sn then ā denotes the reverse ordering of a. For example, in S 7,if a = 05416, then ā = The antidistance d(a, b) between two elements a and b is the distance between ā and b (or between b and a). Woodall [] gave a detailed proof of the two following propositions: Theorem 1 If a S n, then d(a, a) =Z(n). Theorem In a drawing D of K,n on two sets {x, x } and V n, let the clockwise orders in which the edges leave x and x to go to V n be the elements a and b of S n. Then cr D d(a, b), and if n is odd then cr D is of the same parity than d(a, b). the electronic journal of combinatorics 10 (00), #N8
3 Kleitman proved the following equalities: Theorem cr(k 5,n )=4 n/ (n 1)/ (1) cr(k 6,n )=6 n/ (n 1)/. () From this he deduced that cr(k m,n ) (1/5)(m)(m 1) n/ (n 1)/ () in the following way: There are ( ) m K5,n which are subgraphs of K 5 m,n, with the partite with n vertices in the K 5,n being B. Let σ be the sum over all such K 5,n of the number of crossings that each of these K 5,n containinadrawingd. Obviously σ ( ) m 5 Z(5,n). Each crossing appears in exactly ( ) m K5,n. Therefore cr(k m,n ) ( m 5 ) Z(5,n) ( ) m cr(k m,n ) (1/5)(m)(m 1) n/ (n 1)/ We will obtain a small improvement on this lower bound for large values of n, withm 7, by proving that there is a number of K 5,n subgraphs of K m,n which must have more than Z(5,n) crossings in any drawing of K m,n The cyclic ordering of the edges around each b j B can be considered as a cyclic ordering of the elements of A, and therefore as an element of Sm. Suppose we have a K,m which first partite is {u 1,u } and second partite is {u 1,...,u m }. And let c(u 1 ) denote the cyclic ordering of the edges around u 1,andc(u ) denote the cyclic ordering of the edges around u. Woodall [] proved the following theorem: Theorem 4 If a good drawing of K,m has r crossings, there is a sequence Seq(u 1,u ) of r transpositions between adjacent elements in c(u 1 ), such that if we apply this sequence to c(u 1 ) we obtain c(u ), and there is a crossing in the K, subgraph of K,m on vertices u 1,u,u i,u i if and only if exactly one of the transpositions takes place between elements u i and u i in Seq(u 1,u ). (In a good drawing a K, can have one crossing at most.) We can now prove our first lemma: Lemma 1 In any K,7 subgraph of K(m, n) where the two vertices with 7 edges have the same cyclic ordering of edges, there is a K,4 subgraph which has 6 crossings. Proof :Let A beasubsetof7elementsofa, sayw 1,w,...,w 7 (for every l, w l = a k for some k). Let {b k,b l } be one partite of a K,7 subgraph G of K m,n and let A be the second partite. Now suppose c(b k )=c(b l )ing. Let W (b k,b l )bethesetofpairs{w i,w i } of elements of A such that there is a transposition exchanging w i and w i in Seq(b k,b l )inthe drawing of G. Letw y be one element of A,andletA = A \{w y }.LetA 1 be the set of the electronic journal of combinatorics 10 (00), #N8
4 elements a x of A such that {a x,w y } W (b k,b l )anda = A \ A 1. It is clear that every triple {w y,w z,w z } reverses its ordering between c(b k )and c(b l )= c(b k ), which implies that either all three pairs {w z,w z }, {w y,w z }, {w y,w z } belong to W (b k,b l )orexactlyone of these pairs belong to W (b k,b l ). Therefore, if a pair of elements {w z,w z } A either has both of its elements in A 1 or has both of its elements in A,then{w z,w z } W (b k,b l ). Either Card(A ) 4ornot. IfCard(A ) 4, every 4subset of A has 6 subsets that belong W (b k,b l ). If Card(A ) < 4, Card(A 1 ), and every subset of A 1 has  subset belong W (b k,b l ). Let A 1 be such a subset. Then A 1 U{w y} is a 4subset that has 6 subsets in W (b k,b l ). Therefore there exists a subgraph χ of G, having 6 crossings, where one partite is {b k,b l } and the other partite is a 4subset. There are distinct K,5 in G that have χ as a subgraph so each of them must have at least 6 crossings. We will also need the following lemma : Lemma Let D 5,z be an arbitrary drawing of some K 5,z and let t 0 be an element of the partite F with z elements and let T be the set of all elements of F having the same cyclic ordering of edges incident on them as t 0. The elements of T are t 0,t 1,... Let η be the number of pairs {t k T,t k T } such that cr D5,z (t k,t k ) 6 Z(, 5) +. Then Z(5,z)+η cr D5,z. Proof:Let h(t k ) be the sum of the number of crossings of the edges of t k with edges not incident on any vertex of T.Lett min be the element of T such that h(t min ) h(t k ) for all t k T. Using a construction used by Kleitman,[] ( Constructive Argument p19), we can obtain from D 5,z another drawing D 5,z where the position of the edges not incident on a vertex of T remains unchanged, and h(t k )=h(t min ) for all k, and cr D 5,z (t k,t k )=Z(, 5). (The construction mainly consists at placing t k close to t min and letting the edges incident on t k follow the path of the corresponding edges of t min.) Therefore Z(5,z)+η cr D 5,z +η cr D5,z Let θ be the number of distinct K,5 subgraphs of K m,n having at least 6 crossings in D and such that the partite with elements is a subset of B, and the partite with 5 elements is a subset of A, andletσ have the same definition as it had in the proof of (). Then σ ( ) m 5 Z(5,n)+θ. In the following lemma, A is defined in the same way as in the proof of Lemma 1. Lemma Let λ be the number of distinct pairs of elements of B having the same cyclic ordering of edges incident on an element of A. Then, if n 6!, we have ( ) n/6! λ (6!) the electronic journal of combinatorics 10 (00), #N8 4
5 Proof : Let α 1 and α be two distinct elements of (S7)andletn α1 be the number of vertices of B having the cyclic ordering of their edges equal to α 1 and let n α be the number of vertices of B having the cyclic ordering of their edges equal to α.ifn α1 n α, it is always possible to reduce the number of pairs of vertices having the same cyclic ordering of edges by assigning α to one of the vertices which cyclic ordering of edges was α 1.So the minimum possible number of pairs of vertices having the same cyclic ordering of edges can only occur if n α1 n α =1or n α1 n α = 0 for all {α 1,α } S7. Therefore ( ) n/6! λ (6!) Let ν be the minimum number of K,5 subgraphs of a K,7 whose number of crossings must be at least 6 when cyclic ordering of the 7 edges around the vertices of the partite with ) elements are identical. As noted previously ν =. is the number of K,7 subgraphs of K n,m in which a given K,5 appears. ( m 5 θ ( ) m 7 λν ( ) m 5 From the above we can deduce a new lower bound on the crossing number of K m,n. We have: cr(k m,n ) σ ) ( m ( ) m σ Z(5,n)+θ 5 ( ) ( ) ( ) m 5 Z(5,n)+λν m 7 / m 5 cr(k m,n ) ) (4) For sufficiently large m and n, we therefore have : ( m cr(k m,n ) (1/5)m(m 1) n/ (n 1)/ m n Conclusion In this paper, we have proved a new lower bound on the crossing number of K m,n by proving the existence of certain non optimal drawings of K 5,n subgraphs in any drawing of K m,n. By proving the existence of other non optimal drawings, we might perhaps get improvements on the current lower bound. So this method could be one possible way to progress on Zarankiewicz conjecture. the electronic journal of combinatorics 10 (00), #N8 5
6 References [1] R. K. Guy, The decline and fall of Zarankiewicz s theorem, Proof techniques in Graph Theory, F. Harary, ed. Academic Press, New York (1969), [] Daniel J. Kleitman, The crossing number of K 5,n, J. Combinatorial Theory 9 (1970), 15. [] D. R. Woodall, CyclicOrder Graphs and Zarankiewicz s crossing number conjecture, J. Graph Theory 17 (199), [4] R. Bruce Richter, Carsten Thomassen, Relations between crossing numbers of complete and complete bipartite graphs, The American Mathematical Monthly 104 (1997), the electronic journal of combinatorics 10 (00), #N8 6
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