Published: 23 June 2023

Dominator Coloring of Total Graph of Path and Cycle

Minal Shukla1
Foram Chandarana2
1, 2Marwadi University, Rajkot, Gujarat, 360003, India
Corresponding Author:
Minal Shukla
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Abstract

A dominator coloring of a graph G is a proper coloring in which every vertex of G dominates every vertex of at least one-color class possibly its own class and each color class is dominated by at least one vertex. The minimum number of colors required for dominator coloring of G is called the dominator chromatic number of G and is denoted by χdG. In this paper, we have established the relation between dominator chromatic number χdG, chromatic number χG and domination number γG. We have investigated results on total graphs of path and cycle with χdG = χ G + γ G and χdG = χ G + γ G - 1.

Dominator Coloring of Total Graph of Path and Cycle

Highlights

  • Dominator coloring is a variant of proper coloring and it has applications in social networking.
  • Compared the variants of coloring and domination number for total graph of Paths and Cycles.
  • Obtained the two graph classes for total graph of Paths and Cycles.
  • Applied the concept of dominator coloring to interconnection networks.

1. Introduction

Theory of domination and graph coloring are among the most fascinating problems in graph theory, algorithms and combinatorial optimizations. Both the domains are enriched with wide range of research scope in the field of applied sciences. For comprehensive results of coloring and domination in graphs, we refer [1-17] respectively. The domination theory is concerned with a dominating set which requires the minimum set of vertices such that every vertex of a graph not in the set has a neighbor in it while in the concept of graph coloring, it is required to color the vertices using different colors in such a way that both the end vertices of the edge receive different colors. These problems of graph coloring and domination problems are often in relation. These problems are NP hard problems, and they are explored from the point of approximation algorithms [18-22] and parameterized complexity [23-25].

We consider a Graph G=VG,EG as a simple, finite, connected and undirected graph with vertex set VG and edge set EG. A function f:VG1,2,.,nis said to be proper n-coloring of a graph G if fufv for all uvEG, where u and v are adjacent vertices in G. The smallest integer n for which G admits a proper coloring using n colors is known as the chromatic number χG. The color class is the group of vertices that share the same color. The set Nv=uVG: uvEG represents the open neighborhood of vVG whereas the set Nv=Nvv represents the closed neighborhood of v. Every vertex in a graph Gthat is either an element of S or a neighbor of at least one element of S is refereed as a dominating set. The γ-set is called the dominating set with minimal cardinality and its cardinality is the domination number γG. If each vertex of the graph dominates every vertex of a certain color class then the coloring is said to be dominator coloring. The minimum cardinality of colors used in the graph for dominator coloring is called the dominator chromatic number denoted by χdG. It is clear that every vertex dominates its respective color class. The total graph TG of G is the graph with the vertex set VE and two vertices are adjacent whenever they are either adjacent or incident in G.

Gera et al. [26] presented the idea of dominator coloring. Kavitha and David [27-29] determined the dominator chromatic number for several graph families while Arumugam et al. [30] have thoroughly investigated it. Gera [31] has also explored dominator coloring of bipartite graphs. Merouane et al. [32] analyze the dominator chromatic number of different graph families. Vaidya and Shukla [33] have studied the dominator coloring for degree splitting graphs of various graph families. The dominator coloring of some classes of graphs are obtained by Yangyang Zhou et al. [34]. The dominator coloring of some different graphs is explored by T. Manjula et al. [35]. The dominator coloring of Mycielskian graphs is obtained by A. Mohammed Abid and T. R. Ramesh Rao [36]. The dominator coloring of central and middle graph of some special graphs were investigated by J. Arockia Aruldoss and G. Gurulakshmi [37]. Dominator coloring of regular graphs was derived by T. Manjula and R. Rajeswari [38].

The existing State of art for Various results on some domination coloring of graphs can be found in [39-41]. Chellali and Volkmann [42] found relations between the lower domination parameters and the chromatic number of a graph. The concepts of domination and graph coloring are known to be W-complete [43] and para-NP complete respectively in parameterized complexity. The theory of domination and graph coloring have wide number of applications which led to the algorithmic study of numerous variants of these problems. The algorithmic aspects on dominator coloring of graphs are studied by Arumugam et al. [44].

The objective to find a dominator coloring for graph G is to minimize the number of color classes. Here we outline a significant use of a dominator coloring problem in the social network. To draw a graph of social network, vertices are denoted as social players while the connection between them is shown as edges that is two players are joined by an edge if they are friends. Two strangers can become friends by their mutual friends. Also, each friend wants to serve as a significant mediator for other strangers in order to help them establish interpersonal relationships inside the social network. The dominator coloring problem involves finding the minimum groups of players in the social network with the following characteristics:

1) Players in the same group are strangers.

2) Players in the same group can become friends by at least one common mutual friend.

3) Each player is an intermediary of at least one stranger group.

In the existing literature, the various results on particular graph families are obtained for dominator colorings of graph. Also, certain characterizations, bounds and its properties are discussed for the larger graphs. By the observation of existing literature review, we have obtained the dominator coloring of path and cycle by means of the graph operation as total graph and also, we have proved our results in context of domination number and chromatic number of respective graphs.

These results are useful in terms of bounds of dominator coloring for total graph of path and cycle which can be measured using domination number and chromatic number of a graph. By using these results, one can solve the problem of social networks, channel assignment and time table problem.

Using the presented results, we can solve the problems of standard graph families by means of various graph operations in context of proper coloring and domination number. The new research directions are open for researchers to solve the open problems of dominator coloring for different graph families using various graph operations.

The theory of domination and Graph coloring have various applications in the field of communication, computer, social, biological, air traffic flow network and airline scheduling [45].

2. Preliminaries

Proposition 2.1 [46]

1
γTCn=2n5,n0 mod 5,2n5,n0 mod 5.

Proposition 2.2 [47]

For any graph G, χG3 if and only if G has an odd cycle.

Proposition 2.3[48]

For the cycle Cn, χdCn=n2,n=4,n3+1,n=5,n3+2,otherwise.

3. Main Results

Lemma 3.1

2
γTPn=2n-15, n2.

Proof:

Let v1,v2,,vn be the vertices of path Pn and let u1,u2,,un be the newly added vertices corresponding to the edges e1,e2,,en-1 of Pn to obtain TPn. Thus VTPn=v1,v2,.,vn,u1,u2,,un-1. The graph TPn will have 2n-1 vertices and 4n-5 edges. Let S1=vi VTPn:i2mod5 and S2=uiVTPn:i4mod5.

If n0 or 3 mod 5 then S=S1S2 is a dominating set of TPn and S=2n-15.

If n 1 or 2 or 4 mod 5 then S'=S1S2vn is a dominating set of TPn and S'=2n-15. Hence γTPn 2n-15.

Further since γGnΔ+1, it follows that γTPn2n-1Δ+12n-15.

Thus, γTPn=2n-15.

Lemma 3.2

3
χTPn=3, n2

Proof:

Let v1,v2,,vn be the vertices of path Pn and let u1,u2,,un be the newly added vertices corresponding to the edges e1,e2,,en-1 of Pn to obtain TPn.

Thus VTPn=v1,v2,.,vn,u1,u2,,un-1. The graph TPn will have 2n-1 vertices and 4n-5 edges.

Case 1: n=2

The graph TP2, contains an odd cycle. According to Proposition 2.2, χTP23.

By giving the vertices proper coloring as fv1=1, fv2=2, fu1=3.Thus at least three colors are needed for proper coloring. Hence, χTP2=3.

Case 2: n3

By Proposition 2.2, χTPn3 as TPn includes an odd cycle. Assign the proper coloring to the vertices as fvi=1, fui=3 for odd i and fvi=2, fui=4 for even i. Thus, minimum four colors are required for proper coloring. Therefore χTPn=3.

Thus, χTPn=3, n2.

Theorem 3.3

4
χdTPn=χTPn+χTPn-1,n=2,3,4,6,χTPn+χTpn,n5, n6.

Proof:

We keep using the language and notations from Lemma 3.1.

Case 1: n=2, 3, 4, 6

When n = 2

The graph TP2C3, by proposition 2.3, χdTP2=3=χTP2+γTP2-1.

When n = 3

In this case the vertex v2 is the only one vertex in the γ-set of graph TP3. According to Lemma – 3.1, γTP3=1 and by Lemma – 3.2, χTP3=3. Hence, by giving the vertices of the γ-set a number of colors equal to γTP3, we define the optimal coloring for TP3.To the remaining vertices of the graph we give χTP3-1 number of colors. We denote the coloring pattern as fv1=2fv3=3, fv2=1, fu1=3,fu2=2. So every vertex dominates the vertices of at least one color class. A dominator coloring is obtained for the corresponding graphs as a result of this proper coloring. Thus, χdTP3=4=χ[T(P3)+γ[T(P3)]-1.

When n = 4

In this case the set v2,u4 is the only γ-sets of graph TP4. According to Lemma – 3.1, γTP4=2 and by Lemma – 3.2, χTP4=3. Hence, by giving the vertices of the γ-set a number of colors equal to γTP4, we define the optimal coloring for TP4.To the remaining vertices of the graph we give χTP4-1 number of colors. We denote the coloring pattern as fv1=3, fv3=4, fv2=1, fv4=3, fu1=4,fu2=3, fu3=2 so every vertex dominates the vertices of at least one color class. A dominator coloring is obtained for the corresponding graphs as a result of this proper coloring. Thus, χdTP4=4=χ[T(P4)+γ[T(P4)]-1.

When n = 6

In this case the set {v1,v6,u4} or {v2,v5,u3} are only γ-sets of graph T(P6). According to Lemma – 3.1, γT(P6)=3 and by Lemma – 3.2, χ[T(P6)]=4. Allocating a number of colors to the vertices of the γ-set that is equal to γT(P6) in order to determine its optimal coloring. Now we use χ[T(P6)]-1 number of colors to color the remaining vertices.

The coloring pattern can be defined as fv1=fv4=fu2=fu5=3, fv3=1, fv6=fu1=fu4=4,fv2=1,fu4=2,fv5=2,f(u3)=5. Here every vertex dominates the vertices of at least one color class. As a result the proper coloring creates a dominator coloring for the relevant graph. Therefore, χdT(P6)=6=χ[T(P6)+γ[T(P6)]-1.

Case 2: n4, n6

Let S1=viVTPn:i2mod5 and S2= ui VTPn:i4mod5. If n0 or 3 mod 5 then S=S1S2 be a γ-set of TPn. If n1 or 2 or 4 mod 5 then S=S1S2vn be a γ-set of TPn.

We start the coloring process by giving the vertices of the γ-set a number of colors equal to γTPn. For the remaining 2n-γTPn vertices assign χTPn number of colors. All the vertices of at least one-color class are dominated by each vertex. This proper coloring pattern satisfies the condition of dominator coloring. Therefore, χdTPn=χTPn+γTPn.

A dominator coloring of TP6 using six colors is shown in Figure – 1

Fig. 1TP6and its dominator coloring

TP6and its dominator coloring

Lemma 3.5

5
χTCn=3n=34,n4.

Proof:

Let v1,v2,., vn be the vertices of cycle Cn and let u1,u2,,un be the newly added vertices corresponding to the edges e1,e2,..,en of Cn to obtain TCn. Then, VTCn= 2n and ETCn= 4n.

Case 1: For n=3

According to Proposition 2.2, χTC3 3 as the graph TC3 has an odd cycle. To give proper coloring to the nodes fv1=fu3=2,fv2=fu1=1,fv3=fu2=3. Thus, proper coloring is possible with minimum three colors. Hence, χTC3 =3.

Case 2: For n4

Using Proposition – 2.2, χTCn3 as the graph TCn has an odd cycle. To assign proper coloring to the nodes as fvn=3, fvi=1, odd i, fvi=2, even ifun=2, fui=3, odd i, fui=4, even i. By proper coloring at least 4 colors are required. Therefore, χTCn=4.

Thus, χTCn=3,n=3,4,n4.

Theorem 3.6

For n3

6
χdTCn=χTCn+γTCn-1.

Proof:

We continue with the terminology and notations used in Lemma – 3.5.

Case 1: When n=3

The set v1,u3 is one of the γ-set of graph TC3. According to Proposition 2.1, γTC3=2 and by Lemma – 3.5, χTC3=3. Hence, by giving the vertices of the γ-set a number of colors equal to γTC3. We specify the optimal coloring for TC3. Next, we use χTC3-1 no. of colors to color code the remaining vertices. The coloring pattern can be summarized as fv1=1, fv2=fu1=3,fv3=fu2=4,fu3=2. This coloring meets the requirement for dominator coloring. Thus, χdTC3=4.

Case 2: When n=4

The set v1,v3 in this instance is a member of the γ-set of graph TC4. Proposition – 2.1 states that γTC4=2 and Lemma – 3.5 states that χTC4=4. As a result, by giving the vertices of the γ-set a no. of colors equal to γTC4, we specify the ideal coloring for TC4. The remaining vertices are then given colors using χTC4-1 no. of colors fv1=1, fv3=2, fv2=fu1=3, fv4=fu3=4, fu2=fu4=5 are the coloring pattern defined. The dominator coloring requirement is met by this coloring. Therefore, χdTC4=5.

Case 3: When n5

Let S1=viVTCn:i1mod5 and S2= uiVTCn:i3mod5. Then, S=S1 S2 be a γ-sets of TCn.

The coloring process is started by allocating the vertices of the γ-set a no. of colors equal to γTCn. Then using χTCn-1 no. of colors, we apply the colors to the remaining 2n-γTCn vertices because each vertex predominates over at least one color class worth of vertices. For the relevant graphs, this proper coloring pattern results in a dominator coloring. As a result, χdTCn=χTCn+γTCn-1.

A dominator coloring of TC5 using five colors is shown in Figure 2.

Fig. 2TC5 and its dominator coloring

TC5 and its dominator coloring

4. Concluding Remarks

In graph theory, vertex coloring and theory of domination have many applications in computer and communications network such as cell-phone network, vehicular ad hoc network, wireless sensor network etc. The amalgamation of domination and coloring of graphs called the dominator coloring can be used in the study of interconnections networks. Many research papers are published in the era of dominator coloring in which different authors have discussed about the bounds and the dominator chromatic number of standard graph families whereas in our paper, we have obtained the dominator chromatic number for total graph of path and cycle with the help of chromatic number and domination number. That is the characterization of graphs in two classes as χdG=χG+γGand χdG=χG+γG-1.

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About this article

Received
21 February 2023
Accepted
25 April 2023
Published
23 June 2023
Erratum
Value
Authors have identified errors in the paper originally submitted and finally approved (after the acceptance) by the Authors.
For more information read Editor's Note.
Keywords
coloring
domination number
dominator coloring
total graph
Acknowledgements

The authors have not disclosed any funding.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest

The authors declare that they have no conflict of interest.