Bounding the edge cover of a hypergraph

Farhad Shahrokhi

Published 2021-08-18Version 1

Let $H=(V,E)$ be a hypergraph. Let $C\subseteq E$, then $C$ is an {\it edge cover}, or a {\it set cover}, if $\cup_{e\in C} \{v|v\in e\}=V$. A subset of vertices $X$ is {\it independent} in $H,$ if no two vertices in $X$ are in any edge. Let $c(H)$ and $\alpha(H)$ denote the cardinalities of a smallest edge cover and largest independent set in $H$, respectively. We show that $c(H)\le {\hat m}(h)c(H)$, where ${\hat m}(H)$ is a parameter called the {\it mighty degeneracy} of $H$. Furthermore, we show that the inequality is tight and demonstrate the applications in domination theory.