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تسجيل مستخدم جديدPaths, cycles and sprinkling in random hypergraphs
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تأليف
Oliver Cooley
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We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 le j le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard sprinkling argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.
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Given integers $k,j$ with $1le j le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and deter
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ve a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $ell$-cycle if $k-ell$ divides $k$. The remaining values of $ell$ form an interesting open question.
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