Edge-injective and edge-surjective vertex labellings

Stephan Brandt, Jozef Miškuf, Dieter Rautenbach, Friedrich Regent, Imre Z. Ruzsa

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)666-683
Number of pages18
JournalSIAM Journal on Discrete Mathematics
Volume24
Issue number2
DOIs
Publication statusPublished - 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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