Periodica Mathematica Hungarica

, Volume 74, Issue 1, pp 11–21

# Approximate results for rainbow labelings

• Mirka Miller
Article

## 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.

## Keywords

Graph labeling Polynomial method

## Notes

### Acknowledgments

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

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