Double Arithmetic Odd Decomposition [DAOD] of Some Complete 4-Partite Graphs

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 .