### Abstract

For a graph G = (V, E) we consider vertex-k-labellings f : V → {1,2, ,k} for which the induced edge weighting w : E → {2, 3,., 2k} with w(uv) = f(u) + f(v) is injective or surjective or both. We study the relation between these labellings and the number theoretic notions of an additive basis and a Sidon set, present a new construction for a so-called restricted additive basis, and derive the corresponding consequences for the labellings. We prove that a tree of order n and maximum degree δ has a vertex-k-labelling f for which w is bijective if and only if δ ≤ k = n/2. Using this result we prove a recent conjecture of Ivančo and Jendroł concerning edge-irregular total labellings for graphs that are sparse enough.

Original language | English |
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Pages (from-to) | 666-683 |

Number of pages | 18 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 |

### Keywords

- Additive basis
- Edge-irregular total labelling
- Labelling
- Sidon set
- Weak Sidon set
- Weighting

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*24*(2), 666-683. https://doi.org/10.1137/080723065