Local antimagic labeling of graphs

A k-labeling of a graph G is an injective function ϕ from E(G) to m+k real numbers, where m=|E(G)|. Let μG(v)=∑uv∈E(G)ϕ(uv). A graph is called antimagic if G admits a 0-labeling with labels in {1,2,…,|E(G)|} such that μG(u) ≠ μG(v) for any pair u, v ∈ V(G). A well-known conjecture of Hartsfield and Ringel states that every connected graph other than K2 admits an antimagic labeling. Recently, two sets of authors Arumugam, Premalatha, Băca, Semanĭcová-Fen̆ov̆cíková, and Bensmail, Senhaji, Lyngsie independently introduced the weaker notion of a local antimagic labeling, which only distinguishes adjacent vertices by sum with labels in {1,2,…,|E(G)|}. Both sets of authors conjecture that any connected graph other than K2 admits a local antimagic labeling. In this paper, we prove that every subcubic graph without isolated edges admits a local antimagic labeling with |E(G)| positive real labels. We also prove that each graph G without isolated edges admits a local antimagic k-labeling, where k=min{Δ(G)+1,3col(G)+32}, and col(G) is the coloring number of G. Actually, the latter result holds for the list version.