Bounding the edge cover of a hypergraph

TL;DR

This paper establishes bounds on the minimum edge cover size of hypergraphs using the mighty degeneracy parameter, with proofs of tightness and applications in domination theory.

Contribution

It introduces the mighty degeneracy as a new parameter to bound edge covers in hypergraphs, providing tight inequalities and demonstrating applications.

Findings

Bound c(H) by mighty degeneracy and edge cover size

Proved the inequality is tight

Applied results to domination theory

Abstract

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.