Independence in Uniform Linear Triangle-free Hypergraphs

TL;DR

This paper extends classical bounds on the independence number to r-uniform linear triangle-free hypergraphs, providing a new lower bound expressed as a sum involving a recursively defined function.

Contribution

It generalizes Shearer's bound and improves recent results by deriving a new lower bound for the independence number in hypergraphs.

Findings

Established a new lower bound for independence number in hypergraphs.

Generalized Shearer's classical bound to hypergraph setting.

Provided a recursive formula for the bound involving vertex degrees.

Abstract

The independence number $\alpha(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$\alpha(H)\geq \sum_{u\in V(H)}f_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\geq 2$, where \begin{eqnarray*} f_r(0)&=&1\mbox{, and }\\ f_r(d)&=&\frac{1+\Big((r-1)d^2-d\Big)f_r(d-1)}{1+(r-1)d^2}\mbox{ for $d\geq 1$.} \end{eqnarray*}