A note on short cycles in a hypercube |
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Authors: | Maria Axenovich |
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Affiliation: | Department of Mathematics, Iowa State University, Ames, IA 50011, USA |
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Abstract: | How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd?s about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,Cl) be the largest number of edges in a subgraph of a hypercube Qn containing no cycle of length l. It is known that f(n,Cl)=o(|E(Qn)|), when l=4k, k?2 and that . It is an open question to determine f(n,Cl) for l=4k+2, k?2. Here, we give a general upper bound for f(n,Cl) when l=4k+2 and provide a coloring of E(Qn) by four colors containing no induced monochromatic C10. |
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Keywords: | Cycles Hypercube Coloring Ramsey Ramsey property |
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