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arXiv · 2407.08870

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

Zekhaya B. Shozi
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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 $α'_k(G)$. In this paper we study $α_1'(G)$. In particular, we prove a tight lower (resp. upper) bound on $α_1(G)$ 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.

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Zekhaya B. Shozi. 2024-07-11. The $1$-nearly edge independence number of a graph. https://arxiv.org/abs/2407.08870

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