### Signed total $k$-independence in digraphs

#### Abstract

Let $k\ge 2$ be an integer. A function $f:V(D)\rightarrow\{-1,1\}$ defined on the vertex set

$V(D)$ of a digraph $D$ is a signed totalĀ $k$-independence function if $\sum_{x\in N^-(v)}f(x)\le k-1$

for each $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which

arcs go into $v$. The weight of a signed total $k$-independence function $f$ is defined by

$w(f)=\sum_{x\in V(D)}f(x)$. The maximum of weights $w(f)$, taken over all signed total $k$-independence functions

$f$ on $D$, is the signed total $k$-independence number $\alpha_{st}^k(D)$ of $D$.

In this work, we mainly present upper bounds on $\alpha_{st}^k(D)$, as for example

$\alpha_{st}^k(D)\le n-2\lceil(\Delta^-+1-k)/2\rceil$ and

$$\alpha_{st}^k(D)\le\frac{\Delta^++2k-\delta^+-2}{\Delta^++\delta^+}\cdot n,$$

where $n$ is the order, $\Delta^-$ the maximum indegree and $\Delta^+$

and $\delta^+$ are the maximum and minimum outdegree of the digraph $D$. Some of our results imply

well-known properties on the signed total 2-independence number of graphs.

$V(D)$ of a digraph $D$ is a signed totalĀ $k$-independence function if $\sum_{x\in N^-(v)}f(x)\le k-1$

for each $v\in V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which

arcs go into $v$. The weight of a signed total $k$-independence function $f$ is defined by

$w(f)=\sum_{x\in V(D)}f(x)$. The maximum of weights $w(f)$, taken over all signed total $k$-independence functions

$f$ on $D$, is the signed total $k$-independence number $\alpha_{st}^k(D)$ of $D$.

In this work, we mainly present upper bounds on $\alpha_{st}^k(D)$, as for example

$\alpha_{st}^k(D)\le n-2\lceil(\Delta^-+1-k)/2\rceil$ and

$$\alpha_{st}^k(D)\le\frac{\Delta^++2k-\delta^+-2}{\Delta^++\delta^+}\cdot n,$$

where $n$ is the order, $\Delta^-$ the maximum indegree and $\Delta^+$

and $\delta^+$ are the maximum and minimum outdegree of the digraph $D$. Some of our results imply

well-known properties on the signed total 2-independence number of graphs.

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