A Note on Minimum Degree Condition for Hamiltonian $(a,b)$-Cycles in Hypergraphs

Abstract

Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\dots ,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and $B_i\cup A_{i+1}$ (subscripts module $t$) are edges for all $i=0,1,\dots ,t-1$. Let $\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\ge 5k$ and $n$ divisible by $k$. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists $\alpha \in (0,1)$ such that ${\delta }_a(\mathcal{H})\ge (\alpha +o(1))\left(\genfrac{}{}{0ex}{}{n-a}{b}\right)$ and ${\delta }_b(\mathcal{H})\ge (1-\alpha +o(1))\left(\genfrac{}{}{0ex}{}{n-b}{a}\right)$, then $\mathcal{H}$ contains a Hamilton $(a,b)$-cycle. As a corollary, we prove that if ${\delta }_{\ell }(\mathcal{H})\ge (1/2+o(1))\left(\genfrac{}{}{0ex}{}{n-\ell }{k-\ell }\right)$ for some $\ell \ge k/2$, then $\mathcal{H}$ contains a Hamilton $(k-\ell ,\ell )$-cycle and this is asymptotically best possible.

References in corpus (1)

• Hamilton cycles in graphs and hypergraphs: an extremal perspective