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Bounds on the Dynamic Chromatic Number of a Graph in Terms of its Chromatic Number

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A vertex coloring of a graph is called dynamic if the neighborhood of any vertex of degree at least 2 contains at least two vertices of distinct colors. Similarly to the chromatic number χ(G) of a graph G, one can define its dynamic number χd(G) (the minimum number of colors in a dynamic coloring) and dynamic chromatic number χ2(G) (the minimum number of colors in a proper dynamic coloring). We prove that χ2(G) ≤ χ(G) · χd(G) and construct an infinite series of graphs for which this bound on χ2(G) is tight.

For a graph G, set \( k=\left\lceil \frac{2\Delta (G)}{\delta (G)}\right\rceil \) We prove that χ2(G) ≤ (k+1)c. Moreover, in the case where k ≥ 3 and Δ(G) ≥ 3, we prove the stronger bound χ2(G) ≤ kc.

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References

  1. R. L. Brooks, “On coloring the nodes of a network,” Proc. Cambridge Philos. Soc., 37, 194–197 (1941).

    Article  MathSciNet  MATH  Google Scholar 

  2. H.-J. Lui, B. Montgomery, and H. Poon, “Upper bounds of dynamic chromatic number,” Ars Combin., 68, 193–201 (2003).

    MathSciNet  MATH  Google Scholar 

  3. S. Akbari, M. Ghanbari, and S. Jahanbekam, “On the list dynamic coloring of graphs,” Discrete Appl. Math., 157, No. 14, 3005–3007 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Akbari, M. Ghanbari, and S. Jahanbekam, “On the dynamic chromatic number of graphs,” Contemp. Math., 531, 11–18 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. D. V. Karpov, “Dynamic proper colorings of a graph,” Zap. Nauchn. Semin. POMI, 381, 47–77 (2010).

    Google Scholar 

  6. N. V. Gravin and D. V. Karpov, “On proper colorings of hypergraphs,” Zap. Nauchn. Semin. POMI, 391, 79–89 (2011).

    MATH  Google Scholar 

  7. D. Karpov, “An analog of Brooks’ theorem for dynamic colorings,” Moscow J. Combin. Number Theory, 6, No. 1 (2016).

  8. A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari, “On the difference between chromatic number and dynamic chromatic number of graphs,” Discrete Math., 312, No. 17, 2579–2583 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. S.-J. Kim, S. J. Lee, and W.-J Park, “Dynamic coloring and list dynamic coloring of planar graphs,” Discrete Appl. Math., 161, 2207–2212 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    N. Y. Vlasova

  2. St. Petersburg Department of Steklov Institute of Mathematics and St. Petersburg State University, St. Petersburg, Russia

    D. V. Karpov

Authors
  1. N. Y. Vlasova
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  2. D. V. Karpov
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Correspondence to N. Y. Vlasova.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 37–42.

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Vlasova, N.Y., Karpov, D.V. Bounds on the Dynamic Chromatic Number of a Graph in Terms of its Chromatic Number. J Math Sci 232, 21–24 (2018). https://doi.org/10.1007/s10958-018-3855-4

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  • DOI: https://doi.org/10.1007/s10958-018-3855-4

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