Approximate results for rainbow labelings

Abstract

A simple graph $G=(V,\,E)$ is said to be antimagic if there exists a bijection $f{\text {:}}\,E\rightarrow [1,\,|E|]$ 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 $f{\text {:}}\,V\rightarrow [1,\, |V|],$ such that $\forall x,\,y\in V,$

$$\begin{aligned} \sum _{x_i\in N(x)}f\left( x_i\right) \ne \sum _{x_j\in N(y)}f\left( x_j\right) . \end{aligned}$$

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 $[1,\,2n+m-4]$ and, for trees with k inner vertices, in the interval $[1,\,m+k].$ 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 $\Delta$ in the interval $[1,\,n+t(n-t)],$ where $t=\min \{\Delta ,\,\lfloor n/2\rfloor \},$ and, for trees with k leaves, in the interval $[1,\, 3n-4k].$ In particular, all trees with $n=2k$ vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.