On the feedback number of 3-uniform hypergraph

Abstract: Let $H=(V,E)$ be a hypergraph with vertex set $V$ and edge set $E$. $S\subseteq V$ is a feedback vertex set (FVS) of $H$ if $H\setminus S$ has no cycle and $\tau_c(H)$ denote the minimum cardinality of a FVS of $H$. In this paper, we prove $(i)$ if $H$ is a linear $3$-uniform hypergraph with $m$ edges, then $\tau_c(H)\le m/3$. $(ii)$ if $H$ is a $3$-uniform hypergraph with $m$ edges, then $\tau_c(H)\le m/2$ and furthermore, the equality holds on if and only if every component of $H$ is a $2-$cycle. Let $H=(V,E)$ be a hypergraph with vertex set $V$ and edge set $E$. $A\subseteq E$ is a feedback edge set (FES) of $H$ if $H\setminus A$ has no cycle and $\tau_c'(H)$ denote the minimum cardinality of a FES of $H$. In this paper, we prove if $H$ is a $3$-uniform hypergraph with $p$ components, then $\tau_c'(H)\le 2m-n+p$.