Approximate results for rainbow labelings

A simple graph $$G=(V,\,E)$$G=(V,E) is said to be antimagic if there exists a bijection $$f{\text {:}}\,E\rightarrow [1,\,|E|]$$f:E→[1,|E|] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection $$f{\text {:}}\,V\rightarrow [1,\, |V|],$$f:V→[1,|V|], such that $$\forall x,\,y\in V,$$∀x,y∈V,$$\begin{aligned} \sum _{x_i\in N(x)}f\left( x_i\right) \ne \sum _{x_j\in N(y)}f\left( x_j\right) . \end{aligned}$$∑xi∈N(x)fxi≠∑xj∈N(y)fxj.Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval $$[1,\,2n+m-4]$$[1,2n+m-4] and, for trees with k inner vertices, in the interval $$[1,\,m+k].$$[1,m+k]. In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree $$\Delta $$Δ in the interval $$[1,\,n+t(n-t)],$$[1,n+t(n-t)], where $$ t=\min \{\Delta ,\,\lfloor n/2\rfloor \},$$t=min{Δ,⌊n/2⌋}, and, for trees with k leaves, in the interval $$[1,\, 3n-4k].$$[1,3n-4k]. In particular, all trees with $$n=2k$$n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.