Even Star Decomposition of Complete Bipartite Graphs

E Ebin Raja Merly,J Suthiesh Goldy

doi:10.5539/jmr.v8n5p101

Abstract

A decomposition (G1, G2, G3,… , Gn) of a graph G is an Arithmetic Decomposition(AD) if |E(Gi)| = a + (i – 1)d for all i = 1, 2,… , n and a, d∈Z+. Clearly q = n/2 [2a + (n – 1)d]. The AD is a CMD if a = 1 and d = 1. In this paper we introduced the new concept Even Decomposition of graphs. If a = 2 and d = 2 in AD, then q = n(n + 1). That is, the number of edges of G is the sum of first n even numbers 2, 4, 6,… , 2n. Thus we call the AD with a = 2 and d = 2 as Even Decomposition. Since the number of edges of each subgraph of G is even, we denote the Even Decomposition as (G2, G4,… , G2n).

Highlights

In this paper we investigate Even Star decomposition (ESD) of Complete Bipartite Graphs
Proof: Assume K2t,st admits Even Star Decomposition (ESD) (S2, S4, ... , Sk2t+2−2), we know that q(K2t,st) = 2tst, 2tst = n(n+1)
We prove that the result is true for t = g +1, that is to prove K2g+1,sg+1 admits ESD

Summary

Introduction

A decomposition (G1, G2, G3, ,Gn) of G is said to be an Arithmetic Decomposition (AD) if |E(Gi)| = a+(i – 1)d for all i. If a = 1 and d = 2, AD 2 is an Arithmetic Odd Decomposition (AOD). N(n+1) is the sum of first n even numbers 2, 4, 6, , 2n. We call this Decomposition as an Even Decomposition denoted by (G2, G4, G6, , G2n). The following theorem is a necessary and sufficient condition for a graph G admits Even Decomposition

Theorem

Example