A Note on Dominator Chromatic Number of Double Wheel Graph ... · Ä (G ) + Ò (G ) on the...

A Note on Dominator Chromatic Number of Double

Wheel Graph and Friendship Graph

R. Kalaivani

Research Scholar

Department of Mathematics

Kongunadu Arts and Science College

kalaivanirm @yahoo.com

D. Vijayalakshmi

Assistant Professor

Department of Mathematics

Kongunadu Arts and Science College

Coimbatore - 641 029

Abstract

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 color class. In this paper, we establish the dominator chromatic number

for M (Wn,n), C(Wn,n), T (Wn,n), M (F r4 ), C(F r4 ) and T (F r4 ) respectively. n n n

Keywords: Coloring, Domination, Dominator Coloring, Central graph, Middle graph,

Total graph.

Mathematics Subject Classification: 05C15, 05C75.

1 Introduction

For notation and graph theory terminology we general follow [11, 6] . Specif-

ically, let G = (V, E) be a graph with vertex set V and edge set E. The degree of

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International Journal of Pure and Applied MathematicsVolume 119 No. 16 2018, 97-110ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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a vertex v ∈ V is |N (v)| and is denoted dtt(v) (or simply d(v) when the context is

clear). We let δ(G) = min {d(v) : v ∈ V } and ∆(G) = max {d(v) : v ∈ V } denote the

minimum and maximum degrees respectively (if the context is clear, we will simply

let δ = ∆(G) and ∆ = ∆(G). A vertex of degree one is called a pendant, and its

neighbor is called a support vertex.

A set S ⊆ V is a dominating set of G if every vertex in V − S is adjacent to at

least one vertex in S. The domination number γ(G) is the minimum cardinality of

a dominating set of G. A (proper) k-coloring of a graph G is a function c : V (G) →

{1, 2, . . . , k} such that c(x) ƒ= c(y) for any edge xy. The color class ci is the set of

vertices of G that are assigned with color i.

A dominator coloring of a graph G is a proper coloring of G in which every vertex

dominates every vertex of atleast one color class; that is, every vertex in V (G) is

adjacent to all other vertices in its own color class or is adjacent to all vertices from

at least one (other) color class. The concept of a dominator coloring in a graph

was first aforementioned by Gera et al.,[3] and studied further by Gera [4, 5] and

Chellai and Maffray [1]. The upper and lower bounds max {γ(G), χ(G)} ≤ χd(G) ≤

γ(G) + χ(G) on the dominator coloring of graphs in terms of its domination[11]

number and its chromatic number was established by Gera et al. [4, 5].

In the following section we obtain the exact value for χd for Double wheel graph

and Friendship graph.

2 Dominator Chromatic Number of Cycle Re-

lated Graphs

Theorem 2.1. For n ≥ 4, the dominator chromatic number of double wheel graph is,

χd(W

n,n ) =

3 when n is even

4 when n is odd

Proof. Let the vertex set of double wheel graph is defined as, V (Wn,n) = {v0} ∪

International Journal of Pure and Applied Mathematics Special Issue

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{vi : 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n}. A procedure to obtain dominator coloring of dou-

ble wheel graph as follows. Consider the following case.

Case 1. when n is even

We define a coloring function c : V (Wn,n) → {1, 2, 3}. The vertex of Wn,n colored as

follows.

c1 for v0 c(V (Wn,n)) = c2 for vi, ui : 1 ≤ i ≤ n and i is odd

c3 for vi , ui : 1 ≤ i ≤ n and i is even Case 2. when n is odd

c1 for v0

c(V (W

n,n )) =

c2 for vi, ui : 1 ≤ i ≤ n − 2 and i is odd

c3 for vi, ui : 1 ≤ i ≤ n − 1 and i is even

c4 for un, vn.

By the definition of dominator coloring v0 dominates the own color class and vi, ui :

1 ≤ i ≤ n dominates the color class c1. Hence an easy check shows that the dominator

chromatic number of double wheel graph is,

χd(W

n,n ) = 3 when n is even

4 when n is odd

this completes the proof of the theorem.

Theorem 2.2. For n ≥ 8, the dominator chromatic number of Total graph of double

wheel graph is,

χd(T (Wn,n)) = 2n + 1

Proof.

Let V (Wn,n) = {v0} ∪ {vi : 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n}. By the definition of total graph,

V (T (Wn,n)) = {v0 }∪ {vi : 1 ≤ i ≤ n}∪ {ei : 1 ≤ i ≤ n}∪.

ej

: 1 ≤ i ≤ nΣ∪ {ui : 1 ≤ i ≤ n}∪


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c(V (T (Wn,n))) cn+i for ejj : 1 ≤ i ≤ n

odd

.u

j : 1 ≤ i ≤ n

Σ ∪ .

ejj

: 1 ≤ i ≤ nΣ

where ei is the vertex set of T (Wn,n) correspond-

ing to the edge vivi+1 of Wn,n : 1 ≤ i ≤ n − 1 and vnv1 , ej is the vertex set of T (Wn,n)

corresponding to the edge v0vi of Wn,n, 1 ≤ i ≤ n. ejj

is the vertex set of T (Wn,n)

corresponding to the edge v0ui of Wn,n, 1 ≤ i ≤ n. uj

is the vertex set of T (Wn,n) corresponding to the edge uiui+1 of Wn,n, 1 ≤ i ≤ n − 1 and unu1. Since the induced

sub-graphs .

v0 , ej , e

jj : 1 ≤ i ≤ n

Σ forms a clique of order 2n + 1. Now we define a

c : V (T (Wn,n)) → {1, 2, 3, . . . , n, n + 1, . . . , 2n, 2n + 1}. The vertex of V (T (Wn,n))

colored as follows

ci for ej

: 1 i n

i

c2n+1 for v0

The remaining vertices are colored by the following cases

Case 1 when n is even

c2 for ui : 1 ≤ i ≤ n − 1 and i is odd

c4 for ui : 2 ≤ i ≤ n and i is even

i

c(V (T (Wn,n))) =

j

i

cn+2 for vi : 1 ≤ i ≤ n − 1 and i is odd

cn+4 for vi : 2 ≤ i ≤ n and i is even

cn+6 for ei : 1 ≤ i ≤ n − 1 and i is odd

cn+8 for ei : 2 ≤ i ≤ n and i is even


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Case 1 when n is odd

c2 for en, ui : 1 ≤ i ≤ n − 2 and i is odd

c4 for vn, ui : 2 ≤ i ≤ n − 1 and i is even

c6 for uj

: 1 ≤ i ≤ n − 2 and i is odd i

c(V (T (Wn,n))) =

j

i

cn+2 for un, vi : 1 ≤ i ≤ n − 2 and i is odd

cn+4 for uj , vi : 2 ≤ i ≤ n − 1 and i is even

n

cn+6 for ei : 1 ≤ i ≤ n − 2 and i is odd

cn+8 for ei : 2 ≤ i ≤ n − 1 and i is even

By the definition of dominator coloring the vertices v0, ej and e

jj : 1 ≤ i ≤ n

dominates their own color classes and the vertices vi, ui : 1 ≤ i ≤ n dominates

the color class c2n+1. When n is odd, vertices eiei+1 dominates the color class ci+1

where i = 2, 4, 6, 8, . . . , n − 1 and uj , u

j dominates the color class cn+i+1 where

i = 2, 4, 6, 8, . . . , n − 1. When n is even, eiei+1 dominates the color class ci+1

where i = 2, 4, 6, 8, . . . , n − 2 and uj

, uj

dominates the color class cn+i+1 where i = 2, 4, 6, 8, . . . , n − 2. en, e1 dominates the color class c1 and u

j , u

j dominates the

n 1

color class cn+1. Thus χd(T (Wn,n)) ≤ 2n + 1.

On the other hand we cannot assign lesser-than 2n+1 color to V (T (Wn,n)) Because

the induced subgraph v0, ej , e

jj : 1 ≤ i ≤ n forms a clique of order 2n + 1. Hence

i i

χd(T (Wn,n)) = 2n + 1.

Theorem 2.3. For n ≥ 8, the dominator chromatic number of Central graph of

double wheel graph is,

χd(C(Wn,n)) = |4n/3| + 2.

Proof.

Let V (Wn,n) = {v0} ∪ {vi : 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n}. By the definition of central graph,


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c for v

, ,

, n

, ,

, n

V (C(Wn,n)) = {v0}∪{vi : 1 ≤ i ≤ n}∪{ei : 1 ≤ i ≤ n}∪ ej : 1 ≤ i ≤ n ∪{ui : 1 ≤ i ≤ n}∪

.u

j : 1 ≤ i ≤ n

Σ ∪ .

ejj

: 1 ≤ i ≤ nΣ. Now we define a coloring function

c : V (C(Wn,n)) → {1, 2, 3, . . . , n, n + 1, . . . , |4n/3| + 2}. The vertex of V (C(Wn,n))

colored in the following cases

Case 1 : n ≡ 0 mod 3

c2i for v3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

|4n/3|+1 0

c2i−1 for v3i−2 , v3i−1 , 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|2n/3|+2i for u3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|2n/3|+2i−1 for u3i−2 , u3i−1 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫


c2i for v3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|4n/3|+1 for v0

i i

c(V (C(W

n,n ))) = c|2n/3|+2i for u3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|2n/3|+2i−1 for u3i−2 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3|

c|2n/3|+2i−1 for u3i−1 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c2i−1 for v3i−2 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3|

c2i−1 for v3i−1 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫



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, ,

n

c2i for v3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|4n/3|+1 for v0

i i

c(V (C(W

n,n

c2i−1 for v3i−2 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

))) = c2i−1 for v3i−1 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3|

c|2n/3∫+2i−1 for u3i−2 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|2n/3∫+2i−1 for u3i−1 : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3|

c|2n/3∫+2i for u3i : 1 ≤ i ≤ k, 1 ≤ k ≤ |n/3∫

c|4n/3|+2 for un−1 , uj

, ui : 1 ≤ i ≤ n − 3

c|4n/3| for vn−1 , uj

−2

, uj n−1

Hence an easy check shows that χd(C(Wn,n)) = |4n/3| + 2.

Theorem 2.4. For n ≥ 6, the dominator chromatic number of Middle graph of

double wheel graph is,

χd(M (Wn,n)) = 2n + 2

Proof.

Let V (Wn,n) = {v0} ∪ {vi : 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n}. By the definition of middle graph,

V (M (Wn,n)) = {v0}∪{vi : 1 ≤ i ≤ n}∪{ei : 1 ≤ i ≤ n}∪ ej : 1 ≤ i ≤ n ∪{ui : 1 ≤ i ≤ n}∪

.u

j : 1 ≤ i ≤ n

Σ ∪ .

ejj

: 1 ≤ i ≤ nΣ

Now we define a coloring function

c : V (M (Wn,n)) → {1, 2, 3, . . . , n, n + 1, . . . , 2n, 2n + 1, 2n + 2}. The vertex of V (M (Wn,n))

colored as follows

i

c(V (M (Wn,n))) =

jj

i

c2n+1 for v0

c2n+2 for en.


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c for u

j , v : 1 ≤ i ≤ n2n+1

,

odd

The remaining vertices are colored by the following cases

Case 1 when n is even

c2n+1 for ui : 1 ≤ i ≤ n

i

c(V (M (Wn,n))) =

Case 2. when n is even

j

i

c2n+1 for en−1, vi1 ≤ i ≤ n − 2

c1 for ei : 1 ≤ i ≤ n − 3 and i is odd

cn for ei : 2 ≤ i ≤ n − 2 and i is even

c2n+1 for ui : 2 ≤ i ≤ n − 1

c1 for uj

: 2 ≤ i ≤ n − 1 and i is even

i

c(V (M (Wn,n))) = cn for uj

: 1 ≤ i ≤ n − 2 and i is odd

cn for un, ei : 1 ≤ i ≤ n − 2 and i is odd

c1 for u1 , ei : 2 ≤ i ≤ n − 1 and i is even

By the definition of dominator coloring the vertices en, ej : 2 ≤ i ≤ n − 1 and e

jj : 1 ≤

i ≤ n dominates itself.The vertex ui dominates the color class of ejj

: 1 ≤ i ≤ n and

the remaining vertices dominates any one of the color class of ej e

jj : 1 ≤ i ≤ n Hence i i

an easy observation shows that χd(M (Wn,n)) = 2n + 2.

Theorem 2.5. For n ≥ 4, the dominator chromatic number of friendship graph is,

χ (F r(4)) = n + 2 d n

Proof.

Let V (F r(4)) = {v } ∪ {v

: 1 ≤ i ≤ n} ∪ {u

: 1 ≤ i ≤ 2n} be the vertices of friend-

ship graph with 3n + 1 vertices and 4n edges . Let vi : 1 ≤ i ≤ n be the outer


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vertices of F r(4) and ui :: 1 ≤ i ≤ 2n be the inside vertices of F r(4). A procedure

to obtain dominator coloring of F r(4) graph as follows. We define a coloring c :

V (F r(4)) → {c , c , c , . . . , c , c }. n 1 2 3 n+1

n+2

ci for vi : 1 ≤ i ≤ n

cn+2 for v0

It is easy to see that the above define coloring is a dominator coloring with cn+2

colors. By the definition of dominator coloring, vertices v0, vi : 1 ≤ i ≤ n dominates (4)

itself and u : 1 ≤ i ≤ 2n dominates the color class c . Hence χ (F r ) = n + 2.

Theorem 2.6. For n ≥ 4, the dominator chromatic number of Middle graph of

friendship graph is,

χ (M (F r(4))) = 3n + 2

d n

Proof.

Let V (M (F r(4))) = {v }∪ {v : 1 ≤ i ≤ n}∪{u : 1 ≤ i ≤ 2n}∪{e : 1 ≤ i ≤ 2n}∪

.e

j : 1 ≤ i ≤ 2n

Σ be the vertices of Middle graph of friendship graph. Where ei : 1 ≤

i ≤ 2n is the vertex set of M (F r(4)) corresponding to the edge v u : 1 ≤ i ≤ 2n of

F r(4) and e

jj : 1 ≤ i ≤ 2n is the vertex set of M (F r(4)) corresponding to the edge

v : 1 ≤ i ≤ n, u : 1 ≤ i ≤ 2n of F r(4). In M (F r(4)), the vertices induced by the

subgraph of v0, ei : 1 ≤ i ≤ 2n is isomorphic to complete graph of order 2n + 1. A

procedure to obtain dominator coloring of M (F r(4)) graph as follows. We define a

coloring,

for u : 1 ≤ i ≤ 2n

for u : 1 ≤ i ≤ 2n


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even

c : V (M (F r(4))) → {c , c , c , . . . , c , c , c . . . , c , c , c }.

ci for ei : 1 ≤ i ≤ 2n

c2n+i for vi : 1 ≤ i ≤ n

c(V (M ((F r(4)))) = c3n+1 for v0 , ui : 2 ≤ i ≤ 2n and i is even

c3n+1 for ej : 1 ≤ i ≤ 2n and i is odd

i

c3n+2 for ui : 1 ≤ i ≤ 2n and i is odd

Above consignment is a dominator coloring of M (F r(4)) with c

3n+2

colors. In

M (F r(4)), vertex v : 1 ≤ i ≤ n, e : 1 ≤ i ≤ 2n dominates itself and v , u , ei : 1 ≤

i ≤ 2n dominates the color class c . Thus χ (M (F r(4))) ≤ 3n + 2 .

On the other hand we cannot assign lesser-than c

3n+2 colors to M (F r(4)). Since

the sub-graph induced by the vertices v0, ei : 1 ≤ i ≤ 2n forms a clique of order 2n +

1 so we have to assign c2n+1 colors to v0, ei. To satisfies the dominator coloring

definition the vertices vi : 1 ≤ i ≤ n are colored by c2n+i+1 colors. We need at-least one

more new color to color the remaining vertices of M (F r(4)). Hence dominator

coloring with lesser-than c color is not possible thus χ (M (F r(4))) = 3n + 2. 3n+2 d n

Theorem 2.7. For n ≥ 4, the dominator chromatic number of Total graph of


χ (T (F r(4))) = 3n + 1

d n

Proof.

Let V (T (F r(4))) = {v } ∪ {v : 1 ≤ i ≤ n} ∪ {u : 1 ≤ i ≤ 2n} ∪ {e : 1 ≤ i ≤ 2n} ∪

ej : 1 ≤ i ≤ 2n be the vertices of Total graph of friendship graph. Since the vertices

v0, ei : 1 ≤ i ≤ 2n forms a clique of order 2n + 1. To obtain the dominator coloring

of T (F r(4)) with 3n + 1 colors as follows. Now we define a coloring function c :


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v0

V (T (F r(4))) → {c , c , c , . . . , c , c , c . . . , c , c }.

c(V (T ((F r(4)))) =

ci for ei : 1 ≤ i ≤ 2n

c2n+i for vi : 1 ≤ i ≤ n c

for v

c2n for ej : 1 ≤ i ≤ 2n − 1 and i is odd c2n−1

for ej : 2 ≤ i ≤ 2n and i is even c2n−1 for ui

: 1 ≤ i ≤ 2n − 3 and i is odd c2n for ui : 2 ≤ i ≤ 2n − 2 and i is even

c1 for u2n−1 , u2n

The vertices v0, vi : 1 ≤ i ≤ n, dominates their own color classes and ei1 ≤ i ≤ 2n

dominates the color class c3n+1 and uiej

: 1 ≤ i ≤ 2n dominates the color class of

v : 1 ≤ i ≤ n. Hence an easy observation shows that χ (T (F r(4))) = 3n + 1

Theorem 2.8. For n ≥ 4, the dominator chromatic number of Central graph of


χ (C(F r(4))) = 2n + 2 d n

Proof.

Let V (C(F r(4))) = {v } ∪ {v : 1 ≤ i ≤ n} ∪ {u : 1 ≤ i ≤ 2n} ∪ {e : 1 ≤ i ≤ 2n} ∪

ej : 1 ≤ i ≤ 2n be the vertices of Central graph of friendship graph. Since the

vertices vi : 1 ≤ i ≤ n forms a clique of order n. To obtain the dominator coloring

of T (F r(4)) with 2n + 2 colors as follows. Now we define a coloring function c :

V (C(F r(4))) → {c , c , c , . . . , c , c , c . . . , c , c , c }.

ci for vi : 1 ≤ i ≤ n

ci/2 for ui : 2 ≤ i ≤ 2n and i is even

c2n+2 for eiej

: 1 ≤ i ≤ 2n

for u : 1 ≤ i ≤ 2n − 1 and i is odd

for u : 1 ≤ i ≤ 2n − 1 and i is odd


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The vertex vi dominates anyone color class ci : 1 ≤ i ≤ n and the vertex ei : 1 ≤ i ≤

2n dominates the color class c2n+1 and ej

: 1 ≤ i ≤ 2n and i is even, dominates the

color class ci/2 and v0 dominates itself. Next the vertices ej

: 1 ≤ i ≤ 2n − 1 and i is

odd dominates the color class of ui : 1 ≤ i ≤ 2n − 1.Hence an easy check shows that χ (C(F r(4))) = 2n + 2.

d n

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[3] R. Gera, S Horton, C. Rasmussen, Dominator Colorings and Safe Clique Parti-

tions, Congressus Numerantium, (2006), 19-32.

[4] R. Gera, On The Dominator Colorings in Bipartite Graphs in: Proceedings of the

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[5] R.M Gera, On Dominator Coloring in Graphs, Graph Theory Notes N.Y. LII,

(2007), 947 - 952.

[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in

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[8] D. B. West, Introduction to Graph Theorey, 2nd ed., Prentice Hall, USA, 2001.


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[9] A.P. Kazemi, Total Dominator Chromatic Number in Graphs. Manuscript, arXiv:

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latas Mathematica, 103 (2017) 129-137.

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