Linear cycles of given lengths in linear hypergraphs

Abstract: A well-known result of Verstraete states that for each integer k\geq 2 every graph G with average degree at least 8k contains cycles of k consecutive even lengths, the shortest of which is at most twice the radius of G.

In this talk, we extend Verstraete's result for linear cycles in linear r-uniform hypergraphs, where r\geq 3. We show that for each k\geq 2, there exist constants c_1,c_2 depending only on r such that every linear r-graph with average degree at least c_1 k contains linear cycles of k consecutive even lengths and every linear r-graph with average degree at c_2k contains linear cycles of k consecutive lengths. For the even consecutive lengths case, our bound on the shortest cycle length among the cycles obtained is tight, which also yields improved upper bound on the linear Turan number of an even cycle of given length. For the consecutive lengths case, our bound on the shortest cycle length is tight within a constant factor.

The talk will focus on the tools used in establishing the results. We think that these tools can find further applications to other extremal problems on cycles in the hypergraph setting.

This is joint work with Jie Ma and Liana Yepremyan.

combinatorics

Audience: researchers in the topic

Comments: password 061801

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SCMS Combinatorics Seminar

Series comments: Check scmscomb.github.io/ for more information

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