The $1$-nearly edge independence number of a graph

TL;DR

This paper investigates the 1-nearly edge independence number of graphs, establishing bounds and characterizations for graphs with a given number of vertices, and introduces open problems for future research.

Contribution

It provides tight bounds and characterizations for the 1-nearly edge independence number in graphs with specified vertices, advancing understanding of edge independence properties.

Findings

Established tight bounds on (G) for graphs with fixed vertices

Characterized graphs with minimal (G)

Posed open problems for further exploration

Abstract

Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k \subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of $G$ is called the $k$-nearly edge independence number of $G$, and is denoted by $\alpha'_k(G)$. In this paper we study $\alpha_1'(G)$. In particular, we prove a tight lower (resp. upper) bound on $\alpha_1(G)$ if $G$ is a graph with given number of vertices. Furthermore, we present a characterisation of the general (resp. connected) graphs with given number of vertices and smallest $1$-nearly edge independence number. Lastly, we pose an open problem for further exploration of this study.