Domatic number and bichromaticity of a graph

1983 ◽  
pp. 278-285 ◽  
Author(s):  
Bohdan Zelinka
Keyword(s):  
Domatic Number

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

On the Roman {2}-domatic number of graphs

2020 ◽  
pp. 2150052
Author(s):  
A. Giahtazeh ◽  
H. R. Maimani ◽  
A. Iranmanesh
Keyword(s):  
Bipartite Graphs ◽  
Domatic Number ◽  
Join Of Graphs ◽  
Complete Bipartite Graphs ◽  
Appl Math ◽  
Dominating Function

Let [Formula: see text] be a graph. A Roman[Formula: see text]-dominating function [Formula: see text] has the property that for every vertex [Formula: see text] with [Formula: see text], either [Formula: see text] is adjacent to a vertex assigned [Formula: see text] under [Formula: see text], or [Formula: see text] is adjacent to at least two vertices assigned [Formula: see text] under [Formula: see text]. A set [Formula: see text] of distinct Roman [Formula: see text]-dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a Roman[Formula: see text]-domination family (or functions) on [Formula: see text]. The maximum number of functions in a Roman [Formula: see text]-dominating family on [Formula: see text] is the Roman[Formula: see text]-domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we answer two conjectures of Volkman [L. Volkmann, The Roman [Formula: see text]-domatic number of graphs, Discrete Appl. Math. 258 (2019) 235–241] about Roman [Formula: see text]-domatic number of graphs and we study this parameter for join of graphs and complete bipartite graphs.

scholarly journals Upper bounds on the signed total domatic number of graphs

2011 ◽  
Vol 159 (8) ◽  
pp. 832-837 ◽  
Author(s):  
Lutz Volkmann
Keyword(s):  
Upper Bounds ◽  
Domatic Number

scholarly journals An Improved Exact Algorithm for the Domatic Number Problem

2006 ◽  
Author(s):  
T. Riege ◽  
J. Rothe ◽  
H. Spakowski ◽  
M. Yamamoto
Keyword(s):  
Exact Algorithm ◽  
Domatic Number ◽  
Number Problem

Approximating the domatic number

2000 ◽  
Author(s):  
Uriel Feige ◽  
Magnús M. Halldórsson ◽  
Guy Kortsarz
Keyword(s):  
Domatic Number

scholarly journals The double Roman domatic number of a digraph

2020 ◽  
Vol 40 (4) ◽  
pp. 995
Author(s):  
Lutz Volkmann
Keyword(s):  
Domatic Number

VERTEX COVERING TRANSVERSAL DOMATIC NUMBER OF GRAPHS

2017 ◽  
Vol 18 (1) ◽  
pp. 87-105
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Domatic Number ◽  
Vertex Covering

Signed total domatic number of a graph

2008 ◽  
Vol 12 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Mei Guan ◽  
Er-fang Shan
Keyword(s):  
Domatic Number

The Domatic Number Problem in Interval Graphs

1990 ◽  
Vol 3 (4) ◽  
pp. 531-536 ◽  
Author(s):  
Tung-Lin Lu ◽  
Pei-Hsin Ho ◽  
Gerard J. Chang
Keyword(s):  
Interval Graphs ◽  
Domatic Number ◽  
Number Problem
Sign in / Sign up

Export Citation Format

Share Document