Antimagic labeling of forests with sets of consecutive integers

Let G be a graph. We say G is k-shifted antimagic if there exists a bijection f from E(G) to {k+1,…,k+|E(G)|} such that the vertex sum ϕf(v)=∑e∈E(v)f(e) of every vertex v∈V(G) is unique, and is absolutely antimagic if it is k-shifted antimagic for any k∈Z. The ordinary antimagic problem (k=0) of connected graphs was proposed by Hartsfield and Ringel (1990). In this paper, we prove that the P2,P3,P4-free linear forests and the S2-free star forests are absolutely antimagic with only a few exceptions. This extends the results in (Shang, 2016: Shang et al., 2015). Moreover, we prove that the odd tree forests are absolutely antimagic.