Maximizing the number of edges in optimal k-rankings

The Maximum Number of Edges in a Strongly Multiplicative Graph

The number of edges in k-quasi-planar graphs

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound is n(logn) . In the present note, we improve this bound to (n logn)2 c k (n) in the special case where the graph is drawn in ...

Maximizing the Number of Spanning Trees in a Graph W ith n Nodes and m Edges

Consider the class of undirected graphs hav ing 11. nodes and m edges. The proble m to be addressed here is that of finding specific configurations of III edges on the given n nodes so that the resulting graph will contain the largest number of spanning trees. In particular, an explicit solution to this problem will be exhibited for graphs which have "enough" edges. To be specific, let the set ...

the maximum number of edges in a strongly multiplicative graph

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New Lower Bounds for the Number of (<=k)-Edges and the Rectilinear Crossing Number of Kn

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ bn−2 2 c the number of (≤ k)-edges is at least Ek(S) ≥ 3 ( k + 2 2 ) + k ∑ j=b3 c (3j − n + 3), which, for b3 c ≤ k ≤ 0.4864n, improves the previous best lower bound in [11]. As a main consequence, we obtain a new lower bound on the rectilinear crossing numbe...