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# 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} }

- Published 2000 in Journal of Graph Theory
DOI:10.1002/(SICI)1097-0118(200002)33:2%3C66::AID-JGT2%3E3.0.CO;2-L

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