Improved Bound on Vertex Degree Version of Erdős Matching Conjecture

Abstract

For a $k$-uniform hypergraph $H$, let ${\delta }_1(H)$ denote the minimum vertex degree of $H$, and $\nu (H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\ge 3$ and $\beta >0$, there exists an integer $n_0(\beta ,k)$ such that for positive integers $n\ge n_0$ and $m\le (\frac{k}{2(k-1)}-\beta )\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with ${\delta }_1(H)>\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-m}{k-1}\right),$ then $\nu (H)\ge m$. This improves upon earlier results of Bollobás, Daykin and Erdős (1976) for the range $n>2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\ge 3k^{2}m$.

References in corpus (1)

• Fractional and integer matchings in uniform hypergraphs