scholarly journals On Dominator Chromatic Number Of Radial Graph Of Some Graphs

2019 ◽  
Vol 6 (2) ◽  
pp. 1-3
Author(s):  
Kalaivani R ◽  
Vijayalakshmi D
Keyword(s):  
Chromatic Number ◽  
Star Graph ◽  
Color Class ◽  
Radial Graph ◽  
Middle Graph ◽  
Dominator Coloring

A dominator coloring is a coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class. In this paper, we obtain the Dominator Chromatic number of the Radial graph for the Central graph of Star graph, Super-radial graph for Middle graph of Cycle and Central graph of Path.

scholarly journals Power Dominator Chromatic Number for some Special Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 3957-3960
Keyword(s):  
Chromatic Number ◽  
Star Graph ◽  
Proper Coloring ◽  
Induction Method ◽  
Color Class ◽  
Wheel Graph ◽  
Minimum Number ◽  
Fan Graph ◽  
Multiple Edges ◽  
Dominator Coloring

Let G = (V, E) be a finite, connected, undirected with no loops, multiple edges graph. Then the power dominator coloring of G is a proper coloring of G, such that each vertex of G power dominates every vertex of some color class. The minimum number of color classes in a power dominator coloring of the graph, is the power dominator chromatic number . Here we study the power dominator chromatic number for some special graphs such as Bull Graph, Star Graph, Wheel Graph, Helm graph with the help of induction method and Fan Graph. Suitable examples are provided to exemplify the results.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

A note on global dominator coloring of graphs

2021 ◽  
pp. 2150158
Author(s):  
R. Rangarajan ◽  
David. A. Kalarkop
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Even Order ◽  
Dominator Coloring

Global dominator coloring of the graph [Formula: see text] is the proper coloring of [Formula: see text] such that every vertex of [Formula: see text] dominates atleast one color class as well as anti-dominates atleast one color class. The minimum number of colors required for global dominator coloring of [Formula: see text] is called global dominator chromatic number of [Formula: see text] denoted by [Formula: see text]. In this paper, we characterize trees [Formula: see text] of order [Formula: see text] [Formula: see text] such that [Formula: see text] and also establish a strict upper bound for [Formula: see text] for a tree of even order [Formula: see text] [Formula: see text]. We construct some family of graphs [Formula: see text] with [Formula: see text] and prove some results on [Formula: see text]-partitions of [Formula: see text] when [Formula: see text].

Linear time algorithm for dominator chromatic number of trestled graphs

2019 ◽  
Vol 11 (06) ◽  
pp. 1950066
Author(s):  
S. Arumugam ◽  
K. Raja Chandrasekar
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Open Neighborhood ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Color Class ◽  
Minimum Number ◽  
Dominator Coloring

A dominator coloring (respectively, total dominator coloring) of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] such that each closed neighborhood (respectively, open neighborhood) of every vertex of [Formula: see text] contains a color class of [Formula: see text] The minimum number of colors required for a dominator coloring (respectively, total dominator coloring) of [Formula: see text] is called the dominator chromatic number (respectively, total dominator chromatic number) of [Formula: see text] and is denoted by [Formula: see text] (respectively, [Formula: see text]). In this paper, we prove that the dominator coloring problem and the total dominator coloring problem are solvable in linear time for trestled graphs.

scholarly journals On the dominator chromatic number of the generalized caterpillars forest

2020 ◽  
Author(s):  
Soumia AIOULA ◽  
Mustapha CHELLALI ◽  
Noureddine Ikhlef-Eschouf
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Dominator Coloring

A dominator coloring is a proper coloring of the vertices of a graph such that each vertex of the graph dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number of a graph G is the minimum number of color classes in a dominator coloring of G. In this paper, we determine the exact value of the dominator chromatic number of a subclass of forests which we call, generalized caterpillars forest, where every vertex of degree at least three is a support vertex.

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

scholarly journals b-chromatic number of extended corona of some graphs

2020 ◽  
Vol 7 (2) ◽  
pp. 114-117
Author(s):  
Kiruthika S ◽  
Mohanapriya N
Keyword(s):  
Chromatic Number ◽  
Star Graph

In this paper we find out the b-chromatic number for the extended corona of path with complete on the same order Pn.kn path on order n with star graph on order n+1 say ,Pn.kn+1 cycle with complete on the same order ,Cn.kn cycle on order n with star graph on order n+1 say Cn.kn+1, star graph on order n+1 with complete on order n say kn+1.Kn ,complete on order n with star graph on b-coloring, b-chromatic number, extended corona. order n+1 say kn.Kn+1 respectively.

scholarly journals ON DYNAMIC COLORING OF WEB GRAPH

2018 ◽  
Vol 5 (2) ◽  
pp. 11-15
Author(s):  
Aaresh R.R ◽  
Venkatachalam M ◽  
Deepa T
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Web Graph ◽  
Middle Graph

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

Sign in / Sign up

Export Citation Format

Share Document