Abstract
An inclusive distance vertex irregular labeling of a graph G is an assignment of positive integers \(\{1, 2, \ldots , k\}\) to the vertices of G such that for every vertex the sum of numbers assigned to its closed neighborhood is different. The minimum number k for which exists an inclusive distance vertex irregular labeling of G is denoted by \({\widehat{\mathrm{dis}}}(G)\). In this paper we prove that for a simple graph G on n vertices in which no two vertices have the same closed neighborhood \({\widehat{\mathrm{dis}}}(G)\le n^2\).
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Acknowledgements
This work of the first two authors was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education. This work was supported by the Slovak Research and Development Agency under the contract No. APVV-19-0153 and by VEGA 1/0233/18.
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Cichacz, S., Görlich, A. & Semaničová-Feňovčíková, A. Upper Bounds on Inclusive Distance Vertex Irregularity Strength. Graphs and Combinatorics 37, 2713–2721 (2021). https://doi.org/10.1007/s00373-021-02385-8
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DOI: https://doi.org/10.1007/s00373-021-02385-8