H - V -Super-Strong- ( a , d ) -antimagic decomposition of complete bipartite graphs

Abstract

An H -(a,d)-antimagic labeling in a H -decomposable graph G is a bijection f : V ( G ) ∪ E ( G ) → { 1 , 2 , … , p + q } such that ∑ f ( H 1 ) , ∑ f ( H 2 ) , ⋯ , ∑ f ( H h ) forms an arithmetic progression with difference d and first element a . f is said to be H - V -super- ( a , d ) -antimagic if f ( V ( G ) ) = { 1 , 2 , … , p } . Suppose that V ( G ) = U ( G ) ∪ W ( G ) with | U ( G ) | = m and | W ( G ) | = n . Then f is said to be H - V -super-strong- ( a , d ) -antimagic labeling if f ( U ( G ) ) = { 1 , 2 , … , m } and f ( W ( G ) ) = { m + 1 , m + 2 , … , ( m + n = p ) } . A graph that admits a H - V -super-strong- ( a , d ) -antimagic labeling is called a H - V -super-strong- ( a , d ) -antimagic decomposable graph. In this paper, we prove that complete bipartite graphs K m , n are H - V -super-strong- ( a , d ) -antimagic decomposable with both m and n are even.