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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 π΄(ππ§ ) AND M (π«π π) 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. R S. Publication (rspublication.com), [email protected] Page 239 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 ),β= 6 ο· Mini mum degree in, M (Sn ),πΏ=1 ο· Number of vertices in(Dm 3 ),= 5m+1 ο· Maximum degree in (Dm 3 ), β=2m+2 ο· Minimum degree in ( Dm 3 ), δ =2 3. Harmonious coloring of M (ππ§ ) and M ( ππ¦ π ) Theorem 3.1 : For the middle graph of Sunlet graph M(ππ ), the harmonious chromatic number is χH [M (Sn )] = 3n + 1 if n is odd 3n if n is even Proof : Let (ππ ) be the Sunlet graph on ‘2n’ vertices with n pendent edges to the cycle graph Cn . Let π’1 ,π’2 ,π’3 ……..π’π , π£1 ,π£2 ,π£3 ……. π£π 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 (π£π , π£π+1 ),(π’π , π’π+1 ) ,(π’π,1 ) and (π£π , π’π ) R S. Publication (rspublication.com), [email protected] Page 240 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 is sub divided by the vertex π£π,π+1 ,π’π,π+1 ,π’π ,1 πππππ 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 (ππ ) 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 πΊπ and πΊπ R S. Publication (rspublication.com), [email protected] Page 241 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 1 2 3 2 8 5 7 9 6 10 4 3 Fig.2: Middle Graph of Sunlet Graph - M ( πΊπ ) ππ― [M (πΊπ )]= 3n+1 =10 Theorem 3.2: For the Dutch-wind mill graph D3m , ο£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. R S. Publication (rspublication.com), [email protected] Page 242 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 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 οο£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) ππ― (D33)=3n+3=12 R S. Publication (rspublication.com), [email protected] Page 243 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 References: 1. 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