On k-shifted antimagic spider forests

Abstract

Let $G(V,E)$ be a simple graph with $m$ edges. For a given integer $k$, a $k$-shifted antimagic labeling is a bijection $f:E\left(G\right)\to \{k+1,k+2,\dots ,k+m\}$ such that all vertices have different vertex-sums, where the vertex-sum of a vertex $v$ is the sum of the labels assigned to the edges incident to $v$. A graph $G$ is $k$-shifted antimagic if it admits a $k$-shifted antimagic labeling. For the special case when $k=0$, a 0-shifted antimagic labeling is known as antimagic labeling; and $G$ is antimagic if it admits an antimagic labeling. A spider is a tree with exactly one vertex of degree greater than two. A spider forest is a graph where each component is a spider. In this article, we prove that certain spider forests are $k$-shifted antimagic for all $k⩾0$. In addition, we show that for a spider forest $G$ with $m$ edges, there exists a positive integer $k_0<m$ such that $G$ is $k$-shifted antimagic for all $k⩾k_0$ and $k⩽-(m+k_0+1)$.