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Optimal r-dynamic coloring of sparse graphs

Author

Listed:
  • Dan Yi

    (Jiaxing University)

  • Junlei Zhu

    (Jiaxing University)

  • Lixia Feng

    (Jiaxing University)

  • Jiaxin Wang

    (Jiaxing University)

  • Mengyini Yang

    (Jiaxing University)

Abstract

An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least $$min\{d(v),r\}$$ m i n { d ( v ) , r } different classes. The r-dynamic chromatic number ofG, written $$\chi _{r}(G)$$ χ r ( G ) , is the minimum integer k such that G has such a coloring. In this paper, we investigate the r-dynamic $$(r+1)$$ ( r + 1 ) -coloring (i.e. optimal r-dynamic coloring) of sparse graphs and prove that $$\chi _{r}(G)\le r+1$$ χ r ( G ) ≤ r + 1 holds if G is a planar graph with $$g(G)\ge 7$$ g ( G ) ≥ 7 and $$r\ge 16$$ r ≥ 16 , which is a generalization of the case $$r=\Delta $$ r = Δ .

Suggested Citation

  • Dan Yi & Junlei Zhu & Lixia Feng & Jiaxin Wang & Mengyini Yang, 2019. "Optimal r-dynamic coloring of sparse graphs," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 545-555, August.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:2:d:10.1007_s10878-019-00387-0
    DOI: 10.1007/s10878-019-00387-0
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    References listed on IDEAS

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    1. Junlei Zhu & Yuehua Bu & Yun Dai, 2018. "Upper bounds for adjacent vertex-distinguishing edge coloring," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 454-462, February.
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