A Note on the b-Chromatic Number of Corona of Graphs

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550004 ◽  
Author(s):  
P. C. LISNA ◽  
M. S. SUNITHA
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Chromatic Numbers ◽  
Star Graphs ◽  
Color Class ◽  
Wheel Graphs

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 maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

b-Chromatic Sum and b-Continuity Property of Some Graphs

2021 ◽  
pp. 2150012
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Continuity Property ◽  
Proper Coloring ◽  
Star Graphs ◽  
Power Graph ◽  
Color Class ◽  
Wheel Graphs ◽  
Unequal Number ◽  
Chromatic Sum

A b-coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the largest integer [Formula: see text] such that [Formula: see text] has a b-coloring with [Formula: see text] colors. The b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text], is introduced and it is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for any [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. A graph [Formula: see text] is b-continuous, if it admits a b-coloring with [Formula: see text] colors, for every [Formula: see text]. In this paper, the [Formula: see text]-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.

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.

On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

2020 ◽  
Vol 20 (02) ◽  
pp. 2050007
Author(s):  
P. C. LISNA ◽  
M. S. SUNITHA
Keyword(s):  
Image Processing ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Color Class ◽  
Vertex Set ◽  
Chromatic Sum ◽  
Complete Bipartite Graphs

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 φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

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].

Some results on the b-chromatic number in complementary prism graphs

2019 ◽  
Vol 53 (4) ◽  
pp. 1187-1195
Author(s):  
Amel Bendali-Braham ◽  
Noureddine Ikhlef-Eschouf ◽  
Mostafa Blidia
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Complementary Prism

A b-coloring of a graph G is a proper coloring of G with k colors such that each color class has a vertex that is adjacent to at least one vertex of every other color classes. The b-chromatic number is the largest integer k for which G has a b-coloring with k colors. In this paper, we present some results on b-coloring in complementary prism graphs.

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.

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 Rainbow degree-jump coloring of graphs

2021 ◽  
Vol 13 (1) ◽  
pp. 229-239
Author(s):  
E.G. Mphako-Banda ◽  
J. Kok ◽  
S. Naduvath
Keyword(s):  
Chromatic Number ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Proper Coloring ◽  
Color Class ◽  
Necessary And Sufficient

In this paper, we introduce a new notion called the rainbow degree-jump coloring of a graph. For a vertex $v\in V(G)$, let the degree-jump closed neighbourhood of a vertex $v$ be defined as $N_{deg}[v] = \{u:d(v,u)\leq d(v)\}.$ A proper coloring of a graph $G$ is said to be a rainbow degree-jump coloring of $G$ if for all $v$ in $V(G)$, $c(N_{deg}[v])$ contains at least one of each color class. We determine a necessary and sufficient condition for a graph $G$ to permit a rainbow degree-jump coloring. We also determine the rainbow degree-jump chromatic number, denoted by $\chi_{rdj}(G)$, for certain classes of cycle related graphs.

Sign in / Sign up

Export Citation Format

Share Document