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Sharp upper bound of injective coloring of planar graphs with girth at least 5

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Listed:
  • Qiming Fang

    (Tongji University)

  • Li Zhang

    (Tongji University)

Abstract

An injective k-coloring of a graph G is a k-coloring c (not necessarily proper) such that $$c(u)\ne c(v)$$ c ( u ) ≠ c ( v ) whenever u, v has a common neighbor in G. The injective chromatic number of G, denoted by $$\chi _i(G)$$ χ i ( G ) , is the least integer k such that G has an injective k-coloring. We prove that the injective chromatic number of planar graphs with $$g \ge 5$$ g ≥ 5 and $$\Delta \ge 2339$$ Δ ≥ 2339 is at most $$\Delta + 1$$ Δ + 1 , and this bound is sharp.

Suggested Citation

  • Qiming Fang & Li Zhang, 2022. "Sharp upper bound of injective coloring of planar graphs with girth at least 5," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1161-1198, September.
    • Handle: RePEc:spr:jcomop:v:44:y:2022:i:2:d:10.1007_s10878-022-00880-z
    DOI: 10.1007/s10878-022-00880-z
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    References listed on IDEAS

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    1. Wei Dong & Baogang Xu, 2017. "2-Distance coloring of planar graphs with girth 5," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1302-1322, November.
    2. Yuehua Bu & Xubo Zhu, 2012. "An optimal square coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 580-592, November.
    3. Wei Dong & Wensong Lin, 2016. "An improved bound on 2-distance coloring plane graphs with girth 5," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 645-655, August.
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    Cited by:

    1. Qing Cui & Zhenmeng Han, 2024. "Injective edge-coloring of claw-free subcubic graphs," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-32, July.

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