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Group irregularity strength of connected graphs

Author

Listed:
  • Marcin Anholcer

    (Poznań University of Economics
    University of Primorska, UP FAMNIT)

  • Sylwia Cichacz

    (University of Primorska, UP FAMNIT
    AGH University of Science and Technology)

  • Martin Milanic̆

    (University of Primorska, UP FAMNIT
    University of Primorska, UP IAM)

Abstract

We investigate the group irregularity strength ( $$s_g(G)$$ s g ( G ) ) of graphs, that is, we find the minimum value of $$s$$ s such that for any Abelian group $$\mathcal G $$ G of order $$s$$ s , there exists a function $$f:E(G)\rightarrow \mathcal G $$ f : E ( G ) → G such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $$G$$ G of order at least $$3$$ 3 , $$s_g(G)=n$$ s g ( G ) = n if $$n\ne 4k+2$$ n ≠ 4 k + 2 and $$s_g(G)\le n+1$$ s g ( G ) ≤ n + 1 otherwise, except the case of an infinite family of stars. We also prove that the presented labelling algorithm is linear with respect to the order of the graph.

Suggested Citation

  • Marcin Anholcer & Sylwia Cichacz & Martin Milanic̆, 2015. "Group irregularity strength of connected graphs," Journal of Combinatorial Optimization, Springer, vol. 30(1), pages 1-17, July.
    • Handle: RePEc:spr:jcomop:v:30:y:2015:i:1:d:10.1007_s10878-013-9628-6
    DOI: 10.1007/s10878-013-9628-6
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    Cited by:

    1. Anholcer, Marcin & Cichacz, Sylwia, 2019. "Note on the group edge irregularity strength of graphs," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 237-241.
    2. Anholcer, Marcin & Cichacz, Sylwia & Przybyło, Jakub, 2019. "Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 149-155.

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