The crossing number of c₄ × c₄

Abstract

We prove t h a t t h e crossing number of C4 X Ca is 8. 0 1995 John Wiley & Sons, Inc.It has long been conjectured that the crossing number of C, X C,, denoted u(C, X C n ) , is ( m -2)n for m I n . While ( m -2)n is easily seen to be an upper bound (see Figure l), equality has been established only for m = 3,4. Harary, Kainen, and Schwenk [3] proved that u(C3 X C,) = 3, and Beineke and Ringeisen [4] used this to prove that u(C3 X C,) = n for n 2 3 . Beineke and Ringeisen [ l ] also proved that v(C4 X C,) = 2n for n 2 4, but their proof relied on a 1970 result of Eggleton and Guy [2] that u(C4 X C4) = 8. The proof of the latter result has never been published. It is the purpose of this note to prove that u(C4 X C4) = 8.In what follows, all drawings will be assumed to be simple, i.e., no edge crosses itself, adjacent edges do not cross, crossing edges do so only once, edges do not cross vertices, and no more than two edges cross at a common point. A drawing of a graph G is optimal if it has the minimal number of possible crossings; this number is denoted u ( G ) . We note that C4 X Cq is isomorphic with the 4-cube Q4 and that it contains eight subgraphs isomorphic with the 3-cube Q 3 . Definition 1. edges of G that are incident with both X and V(G)\X.For a subset X of vertices of a graph G, 6 X is the set of Lemma 1. then 16x1 2 4.Journal of Graph Theory, Vol. 19, No. 1, 125-129 (1995) Let X be a subset of V = V(Q3). If 1x1 2 2 and IV\Xl 2 2, 0 1995 John Wiley & Sons, Inc.

The crossing numbers of products of 5-vertex graphs with cycles

“…In the opposite case by deleting the vertices of C f 4 and the edges of F ∪I from the graph H ×C 4 we obtain the drawing of the subgraph isomorphic to C 4 × C 4 with at most seven crossings. This is in contradiction with cr(C 4 × C 4 ) = 8 (see [5]). …”

“…Harary et al [8] conjectured that the crossing number of C m × C n is (m − 2)n, for all m, n satisfying 3 m n. Progress toward a possible proof of the conjecture has come slowly. The conjecture has been proved only for m, n satisfying n m, m 7 [1][2][3][4][5]8,15,16,18]. It was recently proved by Glebsky and Salazar [6] that the crossing number of C m × C n equals its long-conjectured value at least for n m(m + 1).…”

The crossing number of P(3,1)×Pn

“…Harary et al [4] showed that cr(C 3 × C 3 ) = 3, and Ringeisen and Beineke [8] showed that cr(C 3 × C n ) = n. Dean and Richter [2] proved that cr(C 4 × C 4 ) = 8, and Beineke and Ringeisen [1] proved that cr(C 4 × C n ) = 2n and cr(K 4 × C n ) = 3n. Let S n−1 and P n be the star and path with n vertices, respectively.…”

On the crossing numbers of Cartesian products with trees

“…. , 7, the base case n = m was published separately [2,3,9,11,25]. The recent state-of-the-art on the problem is the result of Glebsky and Salazar from [10]: cr(C m C n ) = (m − 2)n for m ≥ 3 and n ≥ 1 2 (m + 1)(m + 2).…”