Rainbow numbers for paths in planar graphs
Introduction
In this paper, our graph will be undirected and simple. For the terms not defined here, we refer to [3]. Let be a graph and be the order and size of respectively. For a vertex of the neighborhood and degree of in are denoted by and . Let be any two subsets of and we use and to denote the subgraph of induced on and respectively. We abbreviate to . The join of graphs and written is the graph obtained from by adding the edges . Let . The path and cycle with vertices are denoted by and respectively. Denote by the wheel graph with the central vertex and rim vertices the elements of are its spokes and the elements of are its rim edges.
For a graph under edge-coloring let us call a subgraph of rainbow if for any . Given a family of graphs and a subgraph of let . We call it rainbow number for in regard to . We abbreviate to . When the rainbow number is strongly linked to the anti-Ramsey number which was initialed by Erdös et al [5]. Here . Clearly, . There are several classes of graph for which have been extensively researched [1], [4], [5], [8], [10], [13], [15], [16], [23], [27]. Of which, cliques, paths, cycles and matchings are some of the most popular graph classes, each of which is determined completely step by step.
On the other hand, the researchers studied the rainbow number from another direction. They investigated the problem when the host graph is not complete graphs. All kinds of results on rainbow numbers for cycles, paths and matchings when host graphs are hypergraphs can be found in [6], [9], [24]. When the host graphs are complete bipartite graphs, the rainbow numbers for cycles and mathcings have been determined completely in [22]. All results on rainbow numbers for cycles, paths and matchings when host graphs are planar graphs can be found in ([11], [12], [14], [17], [18], [19], [20], [21], [25]). We suggest looking at the survey [7] for details.
Horňák et al. [11] firstly investigated the rainbow numbers for cycles in plane triangulations. They first showed that for and got some bounds for and . In general, they attained for which were improved by Lan et al. [18]. Likewise, the authors established new upper bounds for and .
The rainbow numbers of cycles in wheel graphs were completely solved by Xu et al. [28]. The authors attained the accurate values of and . For general, the bounds of were obtained, which have been improved by Xu et al. [28]. The main results are described below. Theorem 1.1 [18], [28] For any . For any integer and .
In this paper, we study the rainbow number for paths in planar graphs. Lan et al. [18] firstly attains a lower bound of for . Theorem 1.2 [18] For an integer where . For an integer where . For and (3.1) if then ; (3.2) if then ; (3.3) if then .
Here we organize our paper as follows: in Section 2, the accurate values of for and the bounds of for are attained; and then lower bound of for all are obtained, which improved the results in Theorem 1.2. In Section 3, we determine the accurate value of for and get lower and upper bounds of for all .
Section snippets
Plane triangulations
In this section, we investigate the rainbow number for paths in plane triangulations. For we use to denote the subgraph of obtained by removing any one edge. Let be a plane triangulation with and . The following lemmas are useful. Lemma 2.1 Let be a connected graph of order which contains no copy of . Then . Lemma 2.2 If is a -free planar graph of order then with equality when if ; and with equality when [2]
[17]
Wheel graphs
In this section, we study the rainbow numbers for paths in wheel graphs. Firstly, we will attain the accurate values of when . Theorem 3.1 For . Proof Consider the following -edge-coloring of : all spokes are colored with different colors and all rim edges are colored with new colors arbitrarily. One can see that does not have a rainbow . Thus, . Note that the subgraph obtained from by deleting any two edges contains a
Acknowledgments
We would like to thank Yongtang Shi for helpful discussions. Zhongmei Qin was supported by National Natural Science Foundation of China (No. 11901050) and Natural Science Basic Research Program of Shaanxi (No.2020JQ - 336). Shasha Li was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY18A010002) and the Natural Science Foundation of Ningbo, China (No. 202003N4148). Yongxin Lan was supported by National Natural Science Foundation of China (No. 12001154) and Special
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