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The signed (k,k) -domatic number of digraphs

Seyed Mahmoud Sheikholeslami orcid.org/0000-0003-2298-4744 ; Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
Lutz Volkmann ; Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany

Sažetak

et $D$ be a finite and simple digraph with vertex set $V(D)$, and
let $f:V(D)\rightarrow\{-1,1\}$ be a two-valued function. If $k\ge
1$ is an integer and $\sum_{x\in N^-[v]}f(x)\ge k$ for each $v\in
V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from
which arcs go into $v$, then $f$ is a signed $k$-dominating
function on $D$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signed
$k$-dominating functions on $D$ with the property that
$\sum_{i=1}^df_i(x)\le k$ for each $x\in V(D)$, is called a signed
$(k,k)$-dominating family (of functions) on $D$. The maximum
number of functions in a signed $(k,k)$-dominating family on $D$
is the signed $(k,k)$-domatic number on $D$, denoted by
$d_{S}^{k}(D)$.
In this paper, we initiate the study of the signed $(k,k)$-domatic
number of digraphs, and we present different bounds on
$d_{S}^{k}(D)$. Some of our results are extensions of well-known
properties of the signed domatic number $d_S(D)=d_{S}^{1}(D)$ of
digraphs $D$ as well as the signed $(k,k)$-domatic number
$d_S^k(G)$ of graphs $G$.

Ključne riječi

digraph; signed (k; k)-domatic number; signed k-dominating function; signed k-domination number

Hrčak ID:

93287

URI

https://hrcak.srce.hr/93287

Datum izdavanja:

5.12.2012.

Posjeta: 965 *