Covering and 2-degree-packing numbers in graphs

TL;DR

This paper establishes bounds relating the covering number and a new 2-degree-packing parameter in simple graphs, providing characterizations for graphs that attain these bounds.

Contribution

It introduces the 2-degree-packing number and derives bounds connecting it to the covering number, with characterizations of extremal graphs.

Findings

Bounds:   eta(G)  u_2(G)-1

Characterization of graphs attaining bounds

Relationship holds for connected simple graphs with  > u_2(G)

Abstract

In this paper, we give a relationship between the covering number of a simple graph $G$, $\beta(G)$, and a new parameter associated to $G$ which is called 2-degree-packing number of $G$, $\nu_2(G)$. We prove that$$\lceil \nu_{2}(G)/2\rceil\leq\beta(G)\leq\nu_2(G)-1,$$ for any connected simple graph $G$, with $|E(G)|>\nu_2(G)$, and we give a characterization of simple connected graphs which attains the inequalities.