B7814129219

Double Arithmetic Odd Decomposition [DAOD] of Some Complete 4-Partite Graphs
V. G. Smilin Shali¹, S. Asha^2
1V. G. Smilin Shali*, Research Scholar, Reg No. 12609, Research Department of Mathematics, Nesamony Memorial Christian College, Marthandam, Kanyakumari District, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.
²Dr.S. Asha, Assistant Professor, Research Department of Mathematics, Nesamony Memorial Christian College, Marthandam, Kanyakumari District, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli, Tamil Nadu, India. 
Manuscript received on November 14, 2019. | Revised Manuscript received on 24 November, 2019. | Manuscript published on December 10, 2019. | PP: 3902-3907 | Volume-9 Issue-2, December 2019. | Retrieval Number: B7814129219/2019©BEIESP | DOI: 10.35940/ijitee.B7814.129219
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: Let G be a finite, connected, undirected graph without loops or multiple edges. If G₁ , G₂ , . . . ,Gn are connected edge – disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  . . .  E(Gn), then { G₁ , G₂ , . . . , Gn} is said to be a decomposition of G. The concept of Arithmetic Odd Decomposition [AOD] was introduced by E. Ebin Raja Merly and N. Gnanadhas . A decomposition {G1 , G2 , . . . , Gn } G is said to be Arithmetic Decomposition if each Gi is connected and | E(Gi )| = a+ (i – 1) d , for 1  i  n and a, d  ℤ . When a =1 and d = 2, we call the Arithmetic Decomposition as Arithmetic Odd Decomposition . A decomposition { G1 , G3 , . . . , G2n-1} of G is said to be AOD if | E (Gi ) | = i ,  i = 1, 3, . . . , 2n-1. In this paper, we introduce a new concept called Double Arithmetic Odd Decomposition [DAOD]. A graph G is said to have Double Arithmetic Odd Decomposition [DAOD] if G can be decomposed into 2k subgraphs { 2G1 , 2G3 , . . . , 2G2k-1 } such that each Gi is connected and | E (Gi ) | = i ,  i = 1, 3, . . . , 2k-1. Also we investigate DAOD of some complete 4-partite graphs such as K2,2,2,m , K2,4,4,m and K1 ,2,4,m . 
Keywords: Decomposition of graph, Arithmetic Decomposition, Arithmetic Odd Decomposition [AOD], Double Arithmetic Odd Decomposition [DAOD].
Scope of the Article: Graph Algorithms and Graph Drawing