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

@article{Schelp2000OnQI,
  title={On quadrilaterals in layers of the cube and extremal problems for directed and oriented graphs},
  author={Richard H. Schelp and Andrew Thomason},
  journal={Journal of Graph Theory},
  year={2000},
  volume={33},
  pages={66-82}
}
Erdős has conjectured that every subgraph of the n-cube Qn having more than (1/2+o(1))e(Qn) edges will contain a 4-cycle. In this note we consider ‘layer’ graphs, namely, subgraphs of the cube spanned by the subsets of sizes k − 1, k and k + 1, where we are thinking of the vertices of Qn as being the power set of {1, . . . , n}. Observe that every 4-cycle inQn lies in some layer graph. We investigate the maximum density of 4-cycle free subgraphs of layer graphs, principally the case k = 2. The… CONTINUE READING

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