The rainbow vertex antimagic coloring of tree graphs

Abstract

AbstractLetG(V(G),E(G)) be a connected, simple, and finite graph. Letfbe a bijective function of labeling on graphGfrom the edge setE(G) to natural number up to the number of edges ofG. A rainbow vertex antimagic labeling of graphGis a functionfunder the condition all internal vertices of a pathu–υ, Ɐu, υ∈V(G) have different weight (denoted byw(u)), wherew(u) = ∑uu′∈E(G)f(uu′). IfGhas a rainbow vertex antimagic labeling, then G is a rainbow vertex antimagic coloring, where the every vertex is assigned with the colorw(u). Thervac(G) is a notation of rainbow vertex antimagic connection number of graphGwhich means the minimum colors taken over all rainbow vertex antimagic coloring induced by rainbow vertex antimagic labeling of graphG. The results of this research are the exact value of the rainbow vertex antimagic connection number of star (Sn), double star (DSn), and broom graph (Brn, m).