Antimagic labeling of forests with sets of consecutive integers (Q2065767)

A graph \(G\) is \(k\)-shifted antimagic if there exists a bijection \(f\) from \(E(G)\) to \(\{k + 1, \dots , k + |E(G)|\}\) such that the vertex sum \(\phi_f(v)=\sum_{uv\in E(G)}f(uv)\) of every vertex \(v\in V(G)\) is unique, and is absolutely antimagic if it is \(k\)-shifted antimagic for any \(k\in \mathbb{Z}\). For a given graph \(G\), a graph \(G\) is \(H_1,H_2,\dots,H_s\)-free if none of its components is isomorphic to any \(H_i\). A star \(S_n\) is the complete bipartite graph \(K_{1,n}\) with \(n\geq2\). A graph \(F\) is a linear (or star) forest if every component of \(F\) is a path (or a star). An odd tree forest is a forest with all vertices having odd degrees. It is proved that \(P_2,P_3,P_4\)-free linear forests, \(S_2\)-free star forests and odd tree forests are absolutely antimagic with only a few exceptions.