Periodica Mathematica Hungarica

, Volume 74, Issue 1, pp 11–21 | Cite as

Approximate results for rainbow labelings



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.


Graph labeling Polynomial method 



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


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniv. Politècnica de CatalunyaBarcelonaSpain
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia
  3. 3.Department of MathematicsUniversity of West BohemiaPilsenCzech Republic

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