International Journal of Computer Application Issue 4, Volume 4

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International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
ON HARMONIOUS COLORING OF 𝑴(𝐒𝐧 ) AND M (π‘«π’Ž
πŸ‘)
Mary .U
Associate Professor
Nirmala College for Women
Jothilakshmi . G
Assistant Professor
RVS College of arts and science
Coimbatore
Coimbatore
Abstract : A harmonious coloring is a proper vertex coloring in which every pair
of colors appears on at most one pair of adjacent vertices.
In this paper, harmonious coloring of M (Sn ) and M ( Dm
3 ) are studied . Some
structural properties of them are discussed. Also, their harmonious chromatic
number was obtained .
Keywords : Graph coloring, middle graph, sunlet graph, dutch –windmill graph,
harmonious coloring and harmonious chromatic number.
Classification number:05C75,05C15
1. Introduction
Let G be a finite, undirected graph with no loops and multiple edges. The graph G
has the vertex set V(G) and the edge E(G).Graph coloring is coloring of G such
that no two adjacent vertices share the same color. A harmonious coloring[2, 3,
4, 6] is a proper vertex coloring in which every pair of colors appears on at most
one pair of adjacent vertices. The harmonious chromatic number is the minimum
number of colors needed for any harmonious coloring of G.
The middle graph of G denoted by M(G), is defined as follows :The Vertex set of
M (G) is V(G) ∪E(G) in which two elements are adjacent in M(G) if the following
conditions hold.
(i) x,y ∈ E(G) and they are adjacent in G.
(ii) x ∈V(G), y ∈ E(G) and y is incident on x in G.
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International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
A n – Sunlet graph (Sn ) on 2n vertices is a graph obtained by attaching n pendent
edges to the cycle graph Cn . The dutch-windmill graph( Dm
3 ) ,also called a
friendship graph is obtained by taking m copies of the cycle C3 with a vetex in
common.
2. Structural properties of middle graph of Sunlet Graph and
Dutch-WindmillGraph
ο‚· Number of vertices in M (Sn ),p = 4n
ο‚· Number of edges in M (Sn ), q = 7n
ο‚· Maximum degree in M (Sn ),βˆ†= 6
ο‚· Mini mum degree in, M (Sn ),𝛿=1
ο‚· Number of vertices in(Dm
3 ),= 5m+1
ο‚· Maximum degree in (Dm
3 ), βˆ†=2m+2
ο‚· Minimum degree in ( Dm
3 ), δ =2
3. Harmonious coloring of M (𝐒𝐧 ) and M ( 𝐃𝐦
πŸ‘ )
Theorem 3.1 : For the middle graph of Sunlet graph M(𝑆𝑛 ), the harmonious
chromatic number is
χH [M (Sn )] =
3n + 1 if n is odd
3n
if n is even
Proof : Let (𝑆𝑛 ) be the Sunlet graph on ‘2n’ vertices with n pendent edges to the
cycle graph Cn . Let 𝑒1 ,𝑒2 ,𝑒3 ……..𝑒𝑛 , 𝑣1 ,𝑣2 ,𝑣3 ……. 𝑣𝑛 be the vertices of middle
graph of sunlet graph (taken in clock wise order).
By definition of middle graph, each edge of the Sunlet Graph is subdivided by a
new vertex, assume that each edge (𝑣𝑖 , 𝑣𝑖+1 ),(𝑒𝑖 , 𝑒𝑖+1 ) ,(𝑒𝑛,1 ) and (𝑣𝑗 , 𝑒𝑗 )
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International Journal of Computer Application
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Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
is sub divided by the vertex 𝑣𝑖,𝑖+1 ,𝑒𝑖,𝑖+1 ,𝑒𝑛 ,1 π‘Žπ‘›π‘‘π‘’π‘— for 1≤ i ≤ n, 1≤ j ≤ n
respectively.
Clearly V[M (Sn )]= {vi } ∪ {ui } ∪ {vi,i+1 } ∪ {ui,i+1 } ∪ {un,1 } ∪ {ej }
Now we assign the colors to the vertices of M (𝑆𝑛 ) as follows:
Consider a color class C = {c1, c2 , c3, ……cn, c2n, c2n+1,……………. c3n, c3n+1 }
Case (i): If n is odd
Assign the color ci to vi and ci+ 2n+1 to ui for 1≤ i ≤ n
Assign the color c n+i
+1
to ui,i+1 and color c2n+1 to un,1
Case (ii): If n is even
color ci to vi and color cn+i to ui for 1 ≤ i ≤ n
color c2n+i to ui,i+1 and atlast assign the color c3n to un,1
and ci+1 to ei for 1≤ i ≤n-1
It is clearly verified that the above said coloring is harmonious and is minimum.
Hence χH [M (Sn )] =
3n + 1 if n is odd
3n
if n is even
Fig.1: Sunlet Graph π‘ΊπŸ‘ and π‘ΊπŸ’
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Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
1
2
3
2
8
5
7
9
6
10
4
3
Fig.2: Middle Graph of Sunlet Graph - M ( π‘ΊπŸ‘ )
πŒπ‘― [M (π‘ΊπŸ‘ )]= 3n+1 =10
Theorem 3.2: For the Dutch-wind mill graph D3m , H [M(D3m)] = 3m +3,n ≥3
Proof: Consider the Dutch-wind mill graph D3m formed from m copies of the cycle
C3 with vertices, {ui1, ui2 ,ui3 }, i = 1, 2, 3 … m with the vertex ui1 in common.
(i.e) u11= u11 = u31 …….= um1
Now consider M(D3m), each ui1 ,i = 1, 2, 3 … m, will act as a root vertex. Color
the vertices of M(D3m)as follows.
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Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
Color the root vertex with c1.
The inner subdivision vertices are colored with the color c2 ,c3 ,......c2m+1
(clockwise)
The outer subdivision vertices are colored with c3m-1 ,c3m, c3m+1 (clockwise)
The remaining ‘2m’ actual vertices are colored suitably with colors
3m+(j-1)and,3m+j for 1 ≤j ≤m. It is verified that coloring is harmonious and the
minimum number of colors required for harmonious coloring is 3m + 3.
ο€ οœο£H [M (D3m)] = 3m+3
Fig.3:Dutch - windmill graph D33 and D34
12
9
10
8
10
11
7
6
3
1
5
11
2
4
10
9
Fig.4: Middle graph of Dutch -Windmill graph –M(D33)
πŒπ‘― (D33)=3n+3=12
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International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 4 (July-August 2014)
ISSN: 2250-1797
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