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Rainbow numbers for paths in planar graphs
Applied Mathematics and Computation ( IF 3.5 ) Pub Date : 2021-01-11 , DOI: 10.1016/j.amc.2020.125918
Zhongmei Qin , Shasha Li , Yongxin Lan , Jun Yue

Given a family of graphs F and a subgraph H of FF, 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 k8, 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 4k7. Furthermore, we get lower and upper bounds of rb(Wd,Pk) for all k9 and obtain the accurate values of rb(Wd,Pk) for k{4,5,6,7,8,d+1}.



中文翻译:

平面图中路径的彩虹数

给定一系列图 F 和一个子图 HFF[RbFH 表示最小的数字 ķ 所以有彩虹 H 在任何 ķ-边缘色 F。我们称其为彩虹号H 关于 F。有序所有平面三角剖分的集合ñ 用表示 Ťñ。轮盘图d+1个 和命令的路径 ķ 用表示 w ^dPķ分别。在本文中,我们确定了[RbŤñPķ 对全部 ķ8可以改善[Y. Lan,Shi。和ZX。歌曲,路径和循环的平面反拉姆西数,离散数学。342(11)(2019),3216–3224。]。此外,我们还获得了[RbŤñPķ 对于 4ķ7。此外,我们得到了上下界[Rbw ^dPķ 对全部 ķ9 并获得的准确值 [Rbw ^dPķ 对于 ķ{45678d+1个}

更新日期:2021-01-11
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