Rainbow numbers for paths in planar graphs

摘要

Given a family of graphs and a subgraph of let denote the smallest number so that there is a rainbow in any -edge-colored . We call it rainbow number for in regard to . The set of all plane triangulations of order is denoted by . The wheel graph of order and the path of order are denoted by and respectively. In this paper, we establish lower bounds of for all 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 for . Furthermore, we get lower and upper bounds of for all and obtain the accurate values of for .