Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM)

Vol. 13, Issue 1 (June 2018), pp. 286 - 295

On b-chromatic Number of Graph Families

1Nadeem Ansari, 2R S Chandel and 3Rizwana Jamal

1Department of Mathematics IES, IPS Academy Indore-452001, India [email protected]

2Department of Mathematics Govt. Geetanjali Girls College Bhopal-462001, India [email protected]

3Department of Mathematics Saifia Science College Bhopal-462001, India [email protected]

Received: August 23, 2017; Accepted: March 30, 2018

Abstract

A b-coloring of graph 퐺 is a proper 푘-coloring 퐶 that verifies the following property: for every color class 푐푖, 1 ≤ 푖 ≤ 푘, there exists a vertex 푥푖, with color 푐푖, such that all the other colors in 퐶 are utilized in 푥푖 neighbors. The b-chromatic number of a graph 퐺, denoted by 휑(퐺), is the largest integer 푘 such that 퐺 may have a b-coloring by 푘 colors. In this paper we discuss the b-coloring of prism graph 푌푛, central graph of prism graph 퐶(푌푛), middle graph of prism graph 푀(푌푛) and the total graph of prism graph 푇(푌푛) and we obtain the b-chromatic number for these graphs.

Keywords: b-coloring; prism graph; central graph; middle graph and total graph

AMS-MSC (2010): 05C15, 05C75, 05C76

1. Introduction

Graph coloring is one of the comprehensively researched and popular subject and applied in various disciplines such as channel assignment problems [Sohn (2013)], marketing and micro-economics [Talla Nobibon (2012)], timetabling [Burke et al. (2007), Sabar et al.

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(2012), Badoni and Gupta (2014), Xu et al. (2014)] etc., which are studied by various mathematicians and computer scientists around the world.

Let 퐺 be a finite, undirected graph with no loops and multiple edges with vertex set 푉(퐺) and edge set 퐸(퐺). Definitions not given here may be found in [Harary (1969)]. A 푘-coloring of a graph 퐺 = (푉, 퐸) is defined as a function 푐 on 푉(퐺) = {푣1, 푣2, 푣3, … , 푣푛} into a set of colors 퐶 = {1, 2, … , 푘}, such that 푐(푢) ≠ 푐(푣) for all 푢푣 ∈ 퐸(퐺). The chromatic number of 퐺, denoted by 휒(퐺), is the minimum cardinality 푘 for which 퐺 has a proper 푘-coloring. A dominating proper 푘-coloring of 퐺 is a proper 푘-coloring that verifies the following property: for each color 푖, 1 ≤ 푖 ≤ 푘, there exists a vertex 푥, with color 푖, adjacent to vertices colored with every color 푗, 1 ≤ 푗 ≠ 푖 ≤ 푘. Such vertices are called dominating vertices and the set of vertices {푥1, 푥2, 푥3, … , 푥푛} is called a b-dominating system. The b-chromatic number of 퐺, denoted by 휑(퐺), is the maximum 푘 such that a b-coloring with 푘 colors exist for 퐺.

This coloring was introduced by [Irving and Manlove (1999)]. They proved that determining b-chromatic number 휑(퐺) for an arbitrary graph 퐺 is NP-hard, but polynomial for trees. The following upper bound for the b-chromatic number of a graph, presented in [Irving and Manlove (1999)], has been proved to be very useful. If 퐺 admits a b-coloring with 푚 colors, then 퐺 must have at least m vertices with degree at least 푚 − 1. The 푚-degree of a graph 퐺, denoted by 푚(퐺), is the largest integer 푚 such that 퐺 has m vertices of degree at least 푚 − 1. For a given graph 퐺, it may be easily remarked that 휒 (퐺) ≤ 휑(퐺) ≤ 푚 (퐺). It is an interesting problem to distinguish those graphs 퐺 such that 휑(퐺) = 푚(퐺). From this point of view, d-regular graphs are of special interest. [Kratochvil et al. (2002)] proved that if a d- G has at least 푑4 vertices, then 휑(퐺) = 푑 + 1. Also, [Jakovac and Klavzar (2010)], proved that the b-chromatic number of cubic graphs is 4 except for the petersen graph, 퐾3,3 and the prism over 퐾3. In recent years, there has been vigorous growth of interest in the study of b-coloring since the publication of [Irving and Manlove (1999)]. The b-chromatic number of some special graph classes has been discussed in [Alkhateeb and Kohl (2014), Havet et al. (2012), Kohl (2013), Kouider and Zamime (2017), Lisna and Sunitha (2015), Shaebani (2013), Vivin and Vekatachalam (2015)].

In the present work, we determine the exact value of 휑(퐺), where 퐺 are 푌푛, 퐶(푌푛), 푀(푌푛) and 푇(푌푛) respectively. 2. Preliminaries

The central graph [Immanuel and Gella (2014), Kaliraj et al. (2014), Mahde and Mathad (2016)] of graph 퐺 is obtained by subdividing each edge of 퐺 exactly once and joining all the non adjacent vertices of 퐺.

The middle graph [Mohanapriya et al. (2016), Sutha et al. (2014), Vivin and Vekatachalam (2015)] of 퐺, is defined with the vertex set 푉(퐺) ∪ 퐸(퐺), where two vertices are adjacent, if and only if they are either adjacent edges of 퐺 or one is a vertex and the other is an edge incident with it and it is denoted by 푀(퐺).

Let 퐺 be a graph with vertex set 푉(퐺) and edge set 퐸(퐺). The total graph [Immanuel and Gella (2014), Mohanapriya et al. (2016), Vivin and Vekatachalam (2015)] of 퐺, denoted by 288 Nadeem Ansari et al.

푇(퐺) is defined as follows. The vertex set of 푇(퐺) is 푉(퐺) ∪ 퐸(퐺), where two vertices 푥, 푦 in the vertex set of 푇(퐺) are adjacent in 푇(퐺) in case one the following holds: (i) 푥, 푦 are in 푉(퐺) and 푥 is adjacent to 푦 in 퐺. (ii) 푥, 푦 are in 퐸(퐺) and 푥, 푦 are adjacent in 퐺. (iii) 푥 is in 푉(퐺), 푦 is in 퐸(퐺), and 푥, 푦 are incident in 퐺.

A prism graph 푌푛, is a graph corresponding to the skeleton of 푛-prism. A prism graph 푌푛 has 2푛 vertices and 3푛 edges.

3. Main results

In the main section, we determine the exact value of b-chromatic number of prism graph and its central, middle and total graph.

Theorem 3.1

If 푛 ≥ 4, then the b-chromatic number of prism graph is 휑{Yn} = 4.

Proof:

Let 푌푛 be the prism graph on 2푛 vertices and 3푛 edges. Let 푉(푌푛) = {푣1, 푣2, 푣3, … , 푣푛 } ∪ { 푢1, 푢2, 푢3, … , 푢푛} (taken in order clockwise). We assign 4 colors to the vertices of 푌푛 as follows: Consider color class 퐶 = {푐1, 푐2, 푐3, 푐4}

Assign 푐(푣푖) = 푖, for 1 ≤ 푖 ≤ 4; 푐(푢푖) = 푖 + 2 , for 1 ≤ 푖 ≤ 2; 푐(푢푖+2) = 푖 , for 1 ≤ 푖 ≤ 2.

Now for the remaining vertices we proceed with any proper coloring. We get b-coloring with b-vertices 푢3, 푣2, 푣3 and 푢2 for color classes 1, 2, 3 and 4 respectively. It is maximal since 푌푛 has 푚-degree 4. Thus 휑{Yn} = 4; for 푛 ≥ 4 (see Figure 1).

Figure 1. Prism Graph 푌푛 AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 289

Theorem 3.2

If 푛 ≥ 4, then the b-chromatic number of the central graph of prism graph is 휑{퐶(푌푛)} = 푛 푛 + ⌊ ⌋. 2

Proof:

Let 푉(푌푛) = {푣1, 푣2, 푣3, … , 푣푛 } ∪ { 푢1, 푢2, 푢3, … , 푢푛} and 퐸(푌푛) = {푒푖: 1 ≤ 푖 ≤ 푛 − 1} ∪ ′ ′′ ′ {푒푖 : 1 ≤ 푖 ≤ 푛 − 1} ∪ {푒푖 : 1 ≤ 푖 ≤ 푛} ∪ {푒푛} ∪ {푒푛}, where 푒푖 is the edge 푣푖푣푖+1 (for 1 ≤ ′ ′′ 푖 ≤ 푛 − 1), 푒푖 is the edge 푢푖푢푖+1 (for 1 ≤ 푖 ≤ 푛 − 1), 푒푖 is the edge 푣푖푢푖 (for 1 ≤ 푖 ≤ 푛), 푒푛 ′ is the edge 푣푛푣1 and 푒푛 is the edge 푢푛푢1. By definition of central graph, each edge of the prism graph is subdivided by a new vertex, then 푉(퐶(푌n)) = 푉(푌n) ∪ 퐸(푌n) = {푣i: 1 ≤ 푖 ≤ ′ ′ ′′ ′ 푛} ∪ {푢푖: 1 ≤ 푖 ≤ 푛} ∪ {푣푖 : 1 ≤ 푖 ≤ 푛} ∪ {푢푖: 1 ≤ 푖 ≤ 푛} ∪ {푣푖 : 1 ≤ 푖 ≤ 푛}, where 푣푖 , ′ ′′ ′ ′′ 푢푖 and 푣푖 represents the edge 푒푖, 푒푖 and 푒푖 (1 ≤ 푖 ≤ 푛), respectively.

Figure 2. Central Graph of Prism Graph 퐶(푌푛)

We assign the following coloring to the vertices of 퐶(푌푛) as follows:

푐(푣푖) = 푖 for 1 ≤ 푖 ≤ 푛. ′ ′ 푐(푣1) = 푛, 푐(푣푖+1) = 푖 for 1 ≤ 푖 ≤ 푛 − 1.

290 Nadeem Ansari et al.

If 푛 is odd, then assign the same color to 푢푛 which we assigned to 푣푛 (i.e. 푐푛).

Now assign 푛 푐(푢 ) = 푐(푢 ) = 푛 + 푘 for 1 ≤ 푘 ≤ ⌊ ⌋; 2푘−1 2푘 2 ′′ ′′ 푐(푣푖 ) = 푖 + 1 for 1 ≤ 푖 ≤ 푛 − 1 and 푐(푣푛 ) = 1; ′ ′ 푐(푢푖) = 푖 for 1 ≤ 푖 ≤ 푛 − 1 and 푐(푢푛) = 1.

If we assign distinct colors to the vertices of 푢푖 (for 1 ≤ 푖 ≤ 푛), then it does not produce b- coloring. Also, note that any relocation and combination of the colors to the graph fails to put up the new color. Therefore, this coloring is maximal, so that b-chromatic and has size 푛 + 푛 ⌊ ⌋. Hence, 2

푛 휑(C(Y )) = 푛 + ⌊ ⌋, for 푛 ≥ 4 (see Figure 2). n 2 □

Theorem 3.3

If M(Yn) is the middle graph of prism graph, then 6; 푓표푟 푛 = 3, 4 휑{M(Y )} = { n 7; 푓표푟 푛 ≥ 5

Proof:

Let 푉(푌푛) = {푣1, 푣2, 푣3, … , 푣푛 } ∪ { 푢1, 푢2, 푢3, … , 푢푛} and 퐸(푌푛) = {푒푖: 1 ≤ 푖 ≤ 푛 − 1} ∪ ′ ′′ ′ {푒푖 : 1 ≤ 푖 ≤ 푛 − 1} ∪ {푒푖 : 1 ≤ 푖 ≤ 푛} ∪ {푒푛} ∪ {푒푛}, where 푒푖 is the edge 푣푖푣푖+1 (for 1 ≤ ′ ′′ 푖 ≤ 푛 − 1), 푒푖 is the edge 푢푖푢푖+1 (for 1 ≤ 푖 ≤ 푛 − 1), 푒푖 is the edge 푣푖푢푖 (for 1 ≤ 푖 ≤ 푛), 푒푛 ′ is the edge 푣푛푣1 and 푒푛 is the edge 푢푛푢1. By definition of middle graph, 푉(푀(푌푛)) = ′ ′ 푉(푌푛) ∪ 퐸(푌푛) = {푣푖: 1 ≤ 푖 ≤ 푛} ∪ {푢푖: 1 ≤ 푖 ≤ 푛} ∪ {푣푖 : 1 ≤ 푖 ≤ 푛} ∪ {푢푖: 1 ≤ 푖 ≤ 푛 } ∪ ′′ ′ ′ ′′ ′ ′′ {푣푖 : 1 ≤ 푖 ≤ 푛}, where 푣푖 , 푢푖 and 푣푖 represents the edge 푒푖, 푒푖 and 푒푖 (1 ≤ 푖 ≤ 푛) respectively.

To prove this result we divide the proof into three parts. In the first and second parts we will prove that the b-chromatic number of middle graph of prism graph is 휑{M(Yn)} = 6, if 푛 = 3, 4 and in the third part, we will prove that b-chromatic number of middle graph of prism graph is M(Yn) = 7, if 푛 ≥ 5.

Case I: 푛 = 3. We assign 6 colors, 퐶 = {푐1, 푐2, 푐3, . . . , 푐6} to the vertices of 푀(푌3) as follows:

푐(푣1) = 5; 푐(푣2) = 6 and 푐(푣3) = 4; ′ 푐(푣푖 ) = 푖 for 1 ≤ 푖 ≤ 3; ′′ 푐(푣푖 ) = 3 + 푖 for 1 ≤ 푖 ≤ 3;

푐(푢1) = 6; 푐(푢2) = 4; 푐(푢3) = 5; ′ ′ ′ 푐(푢1) = 2; 푐(푢2) = 3 and 푐(푢3) = 1. AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 291

′ ′′ In 푀(푌3), 푣푖 and 푣푖 (for 1 ≤ 푖 ≤ 3) are b-chromatic vertices for color 푐푖 (for 1 ≤ 푖 ≤ 6).

′ ′ Without loss of generality let 푐(푢1) = 7, then 푣푖 (for 1 ≤ 푖 ≤ 3) cannot be adjacent to a ′ vertex of color 7. So by introducing the new color to 푢1, the coloring is not a b-coloring. Also, note that any relocation and combination of the colors to the graph 푀(푌3) fails to put up the new color. Therefore, this coloring is b-chromatic with maximum coloring possible and has size 6. Thus, 휑(M(Y3)) = 6 (see Figure 3).

Figure 3. Middle Graph of Prism Graph Figure 4. Middle Graph of Prism Graph 푀(푌3) 푀(푌4)

Case II: 푛 = 4. We assign 6 colors, 퐶 = {푐1, 푐2, 푐3, . . . , 푐6} to the vertices of 푀(푌4) as follows:

′ 푐(푣푖 ) = 푖 for 1 ≤ 푖 ≤ 4;

푐(푣푖) = 1 + 푖 for 1 ≤ 푖 ≤ 3; 푐(푣4) = 1; ′′ ′′ 푐(푣1 ) = 푐(푣3 ) = 5; ′′ ′′ 푐 (푣2 ) = 푐 (푣4 ) = 6.

Now assign

푐(푢1) = 푐(푢3) = 6;

푐(푢2) = 푐(푢4) = 5; ′ ′ 푐(푢1) = 4 and 푐(푢푖+1) = 푖 for 1 ≤ 푖 ≤ 3.

′ ′′ ′′ In 푀(푌4); 푣푖 (for 1 ≤ 푖 ≤ 4), 푣1 and 푣2 are b-chromatic vertex. If we use 7 colors to meet ′′ ′ ′′ ′′ the requirement of b-coloring, so let 푐(푣3 ) = 7, then 푣1, 푣1 and 푣2 cannot be adjacent to a ′′ vertex of color 7. So by introducing the new color to 푣3 , the coloring is not a b-coloring. Also, note that any relocation and combination of the colors to the graph 푀(푌4) fails to put up the new color. Therefore, this coloring is b-chromatic with maximum coloring possible and has size 6. Thus, 휑(M(Y4)) = 6 (see Figure 4). 292 Nadeem Ansari et al.

Case III: 푛 ≥ 5. We assign 7 colors to the vertices of 푀(푌푛) as follows: Consider color class 퐶 = {푐1, 푐2, 푐3, . . . , 푐7}.

Assign

푐(푣1) = 푐(푣3) = 6;

푐(푣2) = 푐(푣4) = 7; if 푛 > 5, then assign

푐 (푣5) = 5.

{Special case: If 푛 = 5 (i.e. For 푀(푌5)), then 푐(푣5) = 6}

Now assign

′ ′ 푐 (푣푖 ) = 푖; for 1 ≤ 푖 ≤ 4 and 푐 (푣푛) = 5; ′′ ′′ ′′ 푐 (푣푖 ) = 2 + 푖 for 1 ≤ 푖 ≤ 3; 푐 (푣4 ) = 1 and 푐 (푣푖 ) = 2 for 5 ≤ 푖 ≤ 푛;

푐(푢1) = 푐(푢3) = 1; 푐 (푢2) = 푐 (푢4) = 2 and 푐(푢푖) = 7 for 5 ≤ 푖 ≤ 푛; ′ ′ ′ ′ ′ 푐 (푢1) = 6; 푐 (푢2) = 7; 푐 (푢3) = 3; 푐 (푢푛−1) = 4 and 푐 (푢푛) = 5.

If 푛 = 5, then this coloring is completed. If 푛 > 5, then for the remaining vertices of 푀(푌푛), we further apply the following coloring:

푐(푣푖) = 1 for 6 ≤ 푖 ≤ 푛; 푛 푐(푣′ ) = 6 for 푛 > 5; 3 ≤ 푘 ≤ ⌊ ⌋; 2푘−1 2 푛−1 푐(푣′ ) = 4 for 푛 > 6; 3 ≤ 푘 ≤ ⌊ ⌋; 2푘 2 푛−2 푐(푢′ ) = 5 for n > 5; 2 ≤ 푘 ≤ ⌊ ⌋; 2푘 2 n−1 푐(푢′ ) = 3 for n > 6; 3 ≤ k ≤ ⌊ ⌋. 2푘−1 2

′ ′ ′ ′ 푣푖 (for 1 ≤ 푖 ≤ 4), 푢푛, 푢1 and 푢2 are b-chromatic vertex. Obviously, this coloring is b- chromatic and has size 7. Then 휑{푀(푌푛)} = 7, for 푛 ≥ 5 (see Figure 5). It is maximal since 푀(푌푛) has 푚-degree 7.

Hence, we can conclude from the first, second, and third case,

6; 푓표푟 푛 = 3, 4, 휑{M(Y )} = { □ n 7; 푓표푟 푛 ≥ 5.

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Figure 5. Middle Graph of Prism Graph 푀(푌푛)

Theorem 3.4

If 푛 ≥ 4, then b-chromatic number of total graph 푇(푌푛) of prism graph is 휑{푇(푌푛)} = 7.

Proof:

We get the total graph of graph 퐺 from middle graph by adding edges that join the adjacent vertices of 퐺. We assign 7 colors 퐶 = {푐1, 푐2, 푐3, . . . , 푐7} to the vertices of 푇(푌푛) as follows:

′ ′ 푐(푣푛) = 1, 푐(푢푛) = 3;

푐(푣1) = 2, 푐(푣2) = 5, 푐(푣3) = 1, 푐(푣4) = 7; ′ ′ ′ 푐(푣1) = 7, 푐(푣2) = 6, 푐(푣3) = 4; ′′ ′′ ′′ ′′ 푐(푣1 ) = 4, 푐(푣2 ) = 3, 푐(푣3 ) = 2, 푐(푣4 ) = 2;

푐(푢1) = 5, 푐(푢2) = 4, 푐(푢3) = 3, 푐(푢4) = 6; ′ ′ ′ 푐(푢1) = 6, 푐(푢2) = 7, 푐(푢3) = 5.

These vertices are sufficient to produce a b-chromatic coloring of 푇(푌푛), for 푛 ≥ 4. If 푛 > 4, then for the remaining vertices of 푇(푌푛), we further apply following coloring:

n+1 c(v ) = 4, for n > 4, 3 ≤ k ≤ ⌊ ⌋; 2k−1 2 n 푐(푣 ) = 7, for 푛 > 5, 3 ≤ k ≤ ⌊ ⌋; 2푘 2 푛−1 푐(푣′ ) = 3, for 푛 > 4, 2 ≤ k ≤ ⌊ ⌋; 2푘 2 n 푐(푣′ ) = 6, for 푛 > 5, 3 ≤ k ≤ ⌊ ⌋; 2푘−1 2 294 Nadeem Ansari et al.

′′ c(푣푘 ) = 5, for n > 4, 5 ≤ 푘 ≤ 푛; n+1 푐(푢 ) = 7, for 푛 > 4, 3 ≤ k ≤ ⌊ ⌋; 2푘−1 2 n 푐(푢 ) = 2, for 푛 > 5, 3 ≤ k ≤ ⌊ ⌋; 2푘 2 n−1 푐(푢′ ) = 1, for 푛 > 4, 2 ≤ k ≤ ⌊ ⌋; 2푘 2 n 푐(푢′ ) = 4, for 푛 > 5, 3 ≤ k ≤ ⌊ ⌋. 2푘−1 2

′′ ′′ ′ ′ We get b-coloring with b-vertices 푣3, 푣3 , 푢3, 푣1 , 푣2, 푣2 and 푣1 for color classes 1, 2, 3, 4, 5, 6, and 7, respectively. It is maximal since T(Yn) has 푚-degree 7.

Hence, 휑{푇(푌푛)} = 7, for 푛 ≥ 4 (see Figure 6). □

Figure 6. Total Graph of Prism Graph 푇(푌푛)

4. Conclusion

The study of b-coloring is important due to its applications in many real life problems like clustering, automatic recognition of documents, web service etc. In this paper, we investigated the b-chromatic number of prism graph and its central graph, middle graph and total graph. The investigation of analogous results for different graphs and different operation of above families of graph are still open.

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