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Rainbow numbers for paths in planar graphs

Author

Listed:
  • Qin, Zhongmei
  • Li, Shasha
  • Lan, Yongxin
  • Yue, Jun

Abstract

Given a family of graphs F and a subgraph H of F∈F, let rb(F,H) denote the smallest number k so that there is a rainbow H in any k-edge-colored F. We call it rainbow number for H in regard to F. The set of all plane triangulations of order n is denoted by Tn. The wheel graph of order d+1 and the path of order k are denoted by Wd and Pk, respectively. In this paper, we establish lower bounds of rb(Tn,Pk) for all k≥8, which improves the results in [Y. Lan, Y. Shi and Z-X. Song, Planar anti-Ramsey numbers for paths and cycles, Discrete Math. 342(11) (2019), 3216–3224.]. In addition, we also attain the accurate values or bounds of rb(Tn,Pk) for 4≤k≤7. Furthermore, we get lower and upper bounds of rb(Wd,Pk) for all k≥9 and obtain the accurate values of rb(Wd,Pk) for k∈{4,5,6,7,8,d+1}.

Suggested Citation

  • Qin, Zhongmei & Li, Shasha & Lan, Yongxin & Yue, Jun, 2021. "Rainbow numbers for paths in planar graphs," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    • Handle: RePEc:eee:apmaco:v:397:y:2021:i:c:s0096300320308717
    DOI: 10.1016/j.amc.2020.125918
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    References listed on IDEAS

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    1. Zemin Jin & Yuefang Sun & Sherry H. F. Yan & Yuping Zang, 2017. "Extremal coloring for the anti-Ramsey problem of matchings in complete graphs," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1012-1028, November.
    2. Qin, Zhongmei & Lei, Hui & Li, Shasha, 2020. "Rainbow numbers for small graphs in planar graphs," Applied Mathematics and Computation, Elsevier, vol. 371(C).
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    Cited by:

    1. Ahmed Badawy & Jesus A. Fisteus & Tarek M. Mahmoud & Tarek Abd El-Hafeez, 2021. "Topic Extraction and Interactive Knowledge Graphs for Learning Resources," Sustainability, MDPI, vol. 14(1), pages 1-21, December.

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