On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

on the defining number of (2n - 2)-vertex colorings of kn x kn

On the Biplanar Crossing Number of Kn

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the Euclidean plane. The k-planar crossing number crk(G) of G is min{cr(G1) + cr(G2) + . . .+ cr(Gk)}, where the minimum is taken over all possible decompositions of G into k subgraphs G1, G2, . . . , Gk. The problem of computing the crossing number of complete graphs, cr(Kn), exactly for sm...

The number of 8-cycles in 2-factorizations of Kn

This paper gives a complete solution (with one possible exception) of the problem of constructing 2-factorizations of Kn containing a specified number of 8-cycles.

The linear 3-arboricity of Kn, n and Kn

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, we completely determine lak(G) when G is a balanced complete bipartite graph Kn,n or a complete graph Kn, and k = 3. © 2007 Elsevier B.V. All rights reserved.

The drawing Ramsey number Dr(Kn)

Bounds are determined for the smallest m = Dr(Kn) such that every drawing of Km in the plane (two edges have at most one point in common) contains at least one drawing of Kn with the maximum number (:) of crossings. For n = 5 these bounds are improved to 11 :::; Dr(K5) 113. A drawing D( G) of a graph G is a special realization of G in the plane. The vertices are mapped into different points of ...

More on the crossing number of Kn: Monotone drawings

The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edg...