On the maximum number of edges in a c₄‐free subgraph of q_n

Abstract

For the maximum number f ( n ) of edges in a C4-free subgraph of the n-dimensional cube-graph 0, w e prove f(n) 2 i ( n + f i ) 2 " -' for n = 4f, and f ( n ) 2 i ( n + 0.9,h)2"-' for all n 2 9. This disproves one version of a conjecture of P. Erdos. 0 1995 John Wiley & Sons, Inc.

On quadrilaterals in layers of the cube and extremal problems for directed and oriented graphs

“…In particular, under this the additional assumption the construction of [3] cannot be improved upon. The upper bounds available in general are not as good as this.…”

“…Dejter and Guan [7] gave a subgraph of density around (1 + 3/n)/2. But the best example to date is the elegant construction of Brass, Harborth, and Nienborg [3], which, when n = 4 t , has (n + √ n)2 n−2 edges, or density (1 + 1/ √ n)/2. This example can be modified to provide examples for all values of n, and the examples so derived are known to be best possible for all n ≤ 6 (see Harborth and Nienborg [11]).…”

“…It may well be that this example is extremal for Erdős' conjecture; below is some evidence for this view. The construction of Brass, Harborth, and Nienborg [3] has the attractive feature that it contains at least one edge from every 4-cycle in the cube. It might be imagined that an extremal 4-cycle free subgraph must surely have this feature.…”

“…In turn, they give rise to extremal questions about oriented and unoriented graphs. For example, if k 3 (G) denotes the number of triangles in the graph G, then what is the maximum value of (|G| − 2)e(G) + 2k 3 …”

Turán’s Theorem in the Hypercube

“…The current best upper bound, due to Chung [6], stands at ≈ .623e(Q n ). The best known lower bound is 1 2 (n + √ n) 2 n−1 (for n = 4 r ) due to Brass, Harborth and Nienborg [5].…”