Signed Edge $k$-Independence in Graphs

Abstract

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. If $k\ge 2$ is an integer, then the signed edge $k$-independence function of $G$ is a function $f:E\to \{-1,1\}$ such that $\sum _{e^{\prime }\in N[e]}f(e^{\prime })\le k–1$ for each $e\in E$. The weight of a signed edge $k$-independence function $f$ is $\omega (f)=\sum _{e\in E}f(e).$ The signed edge $k$-independence number ${\alpha }_k^s(G)$ of $G$ is the maximum weight of a signed edge $k$-independence function of $G$. In this paper, we initiate the study of the signed edge $k$-independence number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.