On rainbow antimagic coloring and local edge antimagic coloring of graphs - Tita Khalis Maryati; Fawwaz Fakhrurrozi Hadiputra (Indonesian Combinatorial Society (InaCombS), 2026)
Let G be a connected graph of order n. Let f: V(G) → 1,2,...,n be a bijection and for every uv ∈ E(G) consider wf(uv) = f(u) + f(v) as a coloring of the edge. For a pair of vertices u and v, they are connected by a rainbow path if there exists a path from u to v such that the edges have pairwise distinct colors from wf. The bijection f: V(G) → 1,2,...,n is a rainbow antimagic coloring if for every two vertices there exists a rainbow path. Meanwhile, that bijection f: V(G) → 1,2,...,n is local edge antimagic coloring if every two adjacent edges have distinct weights. The rainbow antimagic connection number rac(G) and local edge antimagic chromatic number χ'lea(G) is the minimum number of distinct edge weights over all rainbow antimagic coloring and local edge antimagic coloring, respectively.We investigate the relationship between rac(G) and χ'lea(G). We prove χ'lea(G) ≤ rac(G) for all graphs and provide conditions for equality. Graphs with diameter at most 2 or satisfying rac(G) = Δ(G) achieve equality. We construct a family Ad with arbitrarily large diameter d where Δ(Ad) = χ'lea(Ad) = rac(Ad), and a family Hn = Kn,n - nK2 of diameter 3 where χ'lea(Hn) = rac(Hn) = 2n-3 > Δ(Hn). These results present hints to the full characterization of graphs G with χ'lea(G) = rac(G).