Weak signed Roman $k$-domatic number of a digraph

Articles in Press

Document Type : Original paper

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RWTH Aachen University, 52056 Aachen, Germany

Abstract

Let $D$ be a digraph with vertex set $V(D)$, and let $k\ge 1$ be an integer. A weak signed Roman $k$-dominating function on a digraph $D$ is a function  $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that $\sum_{u\in N^-[v]}f(u)\ge k$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct weak signed Roman $k$-dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le k$ for each $v\in V(D)$, is called a  weak signed Roman $k$-dominating family (of functions) on $D$. The maximum number of functions in a  weak signed Roman $k$-dominating family on $D$ is the  weak signed Roman $k$-domatic number of $D$, denoted by $d_{wsR}^k(D)$. In this paper we initiate the study of the weak signed Roman $k$-domatic number in digraphs, and we present sharp bounds for $d_{wsR}^k(D)$. In addition, we determine the weak signed Roman $k$-domatic number of some digraphs.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 20 August 2024

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