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Anti-van der Waerden Numbers on Graphs

Abstract

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. A k-term arithmetic progression of a graph G (k-AP) is a list of k distinct vertices such that the distance between consecutive pairs is constant. A rainbow k-AP is a k-AP where each vertex is colored distinctly. This allows for the definition of the anti-van der Waerden number of a graph G, which is the least positive integer r such that every exact r-coloring of G contains a rainbow k-AP. Much of the focus of this paper is on 3-term arithmetic progressions for which general bounds are obtained based on the radius and diameter of a graph. The general bounds are improved for trees and Cartesian products and exact values are determined for some classes of graphs. Longer k-term arithmetic progressions are considered and a connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established.Please confirm if the inserted city and country name for all affiliations is correct. Amend if necessary.The cities and affiliations are correct.

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Acknowledgements

The work of Michael Young is supported in part by the National Science Foundation through Grant 1719841.

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Correspondence to Elizabeth Sprangel.

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Berikkyzy, Z., Schulte, A., Sprangel, E. et al. Anti-van der Waerden Numbers on Graphs. Graphs and Combinatorics 38, 124 (2022). https://doi.org/10.1007/s00373-022-02516-9

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  • DOI: https://doi.org/10.1007/s00373-022-02516-9

Keywords

  • Anti-van der Waerden number
  • Rainbow
  • k-term arithmetic progression
  • Ramsey number