On the dominator chromatic number of the generalized caterpillars forest

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.

A note on global dominator coloring of graphs

10.1142/s1793830921501585 ◽

2021 ◽

pp. 2150158

Author(s):

R. Rangarajan ◽

David. A. Kalarkop

Keyword(s):

Chromatic Number ◽

Upper Bound ◽

Proper Coloring ◽

Color Class ◽

Minimum Number ◽

Even Order ◽

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].

International Journal of Innovative Technology and Exploring Engineering - Special Issue ◽

Power Dominator Chromatic Number for some Special Graphs

10.35940/ijitee.l3466.1081219 ◽

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 ◽

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

10.1142/s1793830919500666 ◽

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 ◽

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.

On strict strong coloring of graphs

10.1142/s1793830921500403 ◽

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 dominator coloring in proper interval graphs and block graphs

10.1142/s1793830915500433 ◽

2015 ◽

Vol 07 (04) ◽

pp. 1550043 ◽

Cited By ~ 2

Author(s):

B. S. Panda ◽

Arti Pandey

Keyword(s):

Chromatic Number ◽

Dominating Set ◽

Domination Number ◽

Maximum Size ◽

Interval Graphs ◽

Proper Coloring ◽

Minimum Cardinality ◽

Block Graphs ◽

Minimum Number ◽

In a graph [Formula: see text], a vertex [Formula: see text] dominates a vertex [Formula: see text] if either [Formula: see text] or [Formula: see text] is adjacent to [Formula: see text]. A subset of vertex set [Formula: see text] that dominates all the vertices of [Formula: see text] is called a dominating set of graph [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text]. A proper coloring of a graph [Formula: see text] is an assignment of colors to the vertices of [Formula: see text] such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of [Formula: see text] is called the chromatic number of [Formula: see text] and is denoted by [Formula: see text]. A dominator coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of [Formula: see text] is called the dominator chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph [Formula: see text] satisfies [Formula: see text], and these bounds are sharp. For a block graph [Formula: see text], where one of the end block is of maximum size, we show that [Formula: see text]. We also characterize the block graphs with an end block of maximum size and attaining the lower bound.

Split and Non-Split Dominator Chromatic Numbers and Related Parameters

Mapana Journal of Sciences ◽

10.12723/mjs.18.5 ◽

2011 ◽

Vol 10 (1) ◽

pp. 52-62

Author(s):

K Kavitha ◽

N G David ◽

N. Selvi

Keyword(s):

Graph Coloring ◽

Proper Coloring ◽

Chromatic Numbers ◽

Color Class ◽

Minimum Number ◽

Dominator Coloring ◽

Cycle Path

A proper graph coloring is defined as coloring the nodes of a graph with the minimum number of colors without any two adjacent nodes having the same color. Dominator coloring of G is a proper coloring in which every vertex of G dominates every vertex of at least one color class. In this paper, new parameters, namely strong split and non-split dominator chromatic numbers and block, cycle, path non-split dominator chromatic numbers are introduced. These parameters are obtained for different classes of graphs and also interesting results are established.

On the double total dominator chromatic number of graphs

10.1142/s1793830921501548 ◽

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.

Introduction to dominated edge chromatic number of a graph

Opuscula Mathematica ◽

10.7494/opmath.2021.41.2.245 ◽

2021 ◽

Vol 41 (2) ◽

pp. 245-257

Author(s):

Mohammad R. Piri ◽

Saeid Alikhani

Keyword(s):

Chromatic Number ◽

Edge Coloring ◽

Color Class ◽

Proper Edge Coloring ◽

Minimum Number

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$, is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $\chi_{dom}^{\prime}(G)$. We obtain some properties of $\chi_{dom}^{\prime}(G)$ and compute it for specific graphs. Also examine the effects on $\chi_{dom}^{\prime}(G)$, when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.

Local chromatic number and topology

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3447 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Gábor Simonyi ◽

Gábor Tardos

Keyword(s):

Lower Bound ◽

Chromatic Number ◽

Topological Method ◽

Proper Coloring ◽

And Topology ◽

Minimum Number ◽

International Audience ◽

Closed Neighborhood

International audience The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.

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

Journal of Interconnection Networks ◽

10.1142/s0219265920500073 ◽

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.