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

TL;DR

This paper establishes minimum degree conditions in hypergraphs that guarantee the existence of Hamiltonian $(a,b)$-cycles, extending classical results and providing asymptotically optimal thresholds.

Contribution

It introduces new degree conditions involving intersection concentration inequalities that ensure Hamilton $(a,b)$-cycles in hypergraphs, generalizing previous theorems.

Findings

Degree conditions guarantee Hamilton $(a,b)$-cycles

Asymptotically optimal thresholds for certain degree parameters

Application of concentration inequalities in hypergraph cycle existence

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 $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ 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 $i=0,1,\ldots,t-1$. Let $\mathcal{H}$ be a $k$-uniform $n$-vertex hypergraph with $n\geq 5k$ 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 $\delta_a(\mathcal{H})\geq (\alpha +o(1))\binom{n-a}{b}$ and $\delta_b(\mathcal{H})\geq (1-\alpha +o(1))\binom{n-b}{a}$, then $\mathcal{H}$ contains a Hamilton $(a,b)$-cycle. As a corollary, we prove that if $\delta_{\ell}(\mathcal{H})\geq (1/2 +o(1))\binom{n-\ell}{k-\ell}$ for some $\ell \geq k/2$, then $\mathcal{H}$ contains a Hamilton $(k-\ell,\ell)$-cycle and this is asymptotically best possible.