Linear Forest mP 3 Plus a Longer Path P n Becoming Antimagic

An edge labeling of a graph G is a bijection f from E ( G ) to a set of | E ( G ) | integers. For a vertex u in G , the induced vertex sum of u , denoted by f + ( u ) , is defined as f + ( u ) = ∑ u v ∈ E ( G ) f ( u v ) . Graph G is said to be antimagic if it has an edge labeling g such that g ( E ( G ) ) = { 1 , 2 , ⋯ , | E ( G ) | } and g + ( u ) ≠ g + ( v ) for any pair u , v ∈ V ( G ) with u ≠ v . A linear forest is a union of disjoint paths of orders greater than one. Let m P k denote a linear forest consisting of m disjoint copies of path P k . It is known that m P 3 is antimagic if and only if m = 1 . In this study, we add a disjoint path P n ( n ≥ 4 ) to m P 3 and develop a necessary condition and a sufficient condition whereby the new linear forest m P 3 ⋃ P n may be antimagic.