# Approximate results for rainbow labelings

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## 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 Using the polynomial method of Alon we prove that there are antimagic injections of any graph

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

*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.## Keywords

Graph labeling Polynomial method## Notes

### Acknowledgments

We are grateful to one of the referees for helpful comments and suggestions.

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© Akadémiai Kiadó, Budapest, Hungary 2016