Post on 18-Aug-2020
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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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} ∪
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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∫
Case 2 : n ≡ 1 mod 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∫
Case 3 : n ≡ 2 mod 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
friendship graph is,
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
friendship graph is,
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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