Rainbow numbers for paths in planar graphs

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Abstract

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}.

Introduction

In this paper, our graph will be undirected and simple. For the terms not defined here, we refer to [3]. Let F=(V(F),E(F)) be a graph and |F|, |E(F)| be the order and size of F, respectively. For a vertex x of F, the neighborhood and degree of x in F are denoted by NF(x) and dF(x). Let X,Y be any two subsets of V(F) and YX:=YX, we use F[X] and FX to denote the subgraph of F induced on X and V(F)X, respectively. We abbreviate F{x} to Fx. The join of graphs F and H, written F+H, is the graph obtained from FH by adding the edges {xy:xV(F),yV(H)}. Let Tn={T:Tisaplanetriangulationand|T|=n}. The path and cycle with k vertices are denoted by Pk and Ck, respectively. Denote by Wd=K1+Cd the wheel graph with the central vertex v and rim vertices v1,v2,,vd, the elements of {si=vvi:i=1,,d} are its spokes and the elements of {ri=vivi+1:i=1,,d(modd)} are its rim edges.

For a graph F under edge-coloring f, let us call a subgraph H of F rainbow if f(e)f(e) for any e,eE(H). Given a family of graphs F and a subgraph H of FF, let rb(F,H)=min{k:thereisarainbowHinanyk-edge-coloredF}. We call it rainbow number for H in regard to F. We abbreviate rb({F},H) to rb(F,H). When F={Kn}, the rainbow number is strongly linked to the anti-Ramsey number f(Kn,H), which was initialed by Erdös et al [5]. Here f(Kn,H)=max{k:thereexistssomek-edge-coloringofKnwithoutrainbowH}. Clearly, rb(Kn,H)=f(Kn,H)+1. 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 F 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 rb(Tn,C3)=3n/22 for n4 and got some bounds for C4 and C5. In general, they attained rb(Tn,Ck)3(k3)n/(k2)(8k25)/(k2) for 6kn, which were improved by Lan et al. [18]. Likewise, the authors established new upper bounds for C6 and C7.

The rainbow numbers of cycles in wheel graphs were completely solved by Xu et al. [28]. The authors attained the accurate values of rb(Wd,C4) and rb(Wd,C5). For general, the bounds of rb(Wd,Ck) were obtained, which have been improved by Xu et al. [28]. The main results are described below.

Theorem 1.1 [18], [28]

  • (1)

    For any d3, rb(Wd,C4)=4d3+1.

  • (2)

    For any integer k5 and dk1, rb(Wd,Ck)=2k7k3d+1.

In this paper, we study the rainbow number for paths in planar graphs. Lan et al. [18] firstly attains a lower bound of rb(Tn,Pk) for nk8.

Theorem 1.2 [18]

  • (1)

    For an integer n8, rb(Tn,P8)(3nγ1)/2, where γ(n+1)mod2.

  • (2)

    For an integer n9, rb(Tn,P9)(3nγ+2)/2, where γ(n+2)mod2.

  • (3)

    For nk10 and γkmod2, (3.1) if kn<3k/2+γ5, then rb(Tn,Pk)n+2k11; (3.2) if 3k/2+γ5n5k/2+γ15, then rb(Tn,Pk)(3n+9k/2+3γ41)/2; (3.3) if n>5k/2+γ15, then rb(Tn,Pk)2n+k13.

Here we organize our paper as follows: in Section 2, the accurate values of rb(Tn,Pk) for k{5,7} and the bounds of rb(Tn,Pk) for k{4,6} are attained; and then lower bound of rb(Tn,Pk) for all k8 are obtained, which improved the results in Theorem 1.2. In Section 3, we determine the accurate value of rb(Wd,Pk) for k{4,5,6,7,8,d+1} and get lower and upper bounds of rb(Wd,Pk) for all k9.

Section snippets

Plane triangulations

In this section, we investigate the rainbow number for paths in plane triangulations. For TTn, we use T to denote the subgraph of T obtained by removing any one edge. Let T*=K2+Pn2 be a plane triangulation with V(K2)={v1,v2} and Pn2=w1w2wn3wn2. The following lemmas are useful.

Lemma 2.1

[2]

Let G be a connected graph of order n which contains no copy of P5. Then |E(G)|n.

Lemma 2.2

[17]

If G is a P6-free planar graph of order n6, then e(G)2n2 with equality when G=2K5 if n=10; and e(G)2n3 with equality when G=K2+K

Wheel graphs

In this section, we study the rainbow numbers for paths in wheel graphs. Firstly, we will attain the accurate values of rb(Wd,Pk) when k{d+1,4,5,6,7,8}.

Theorem 3.1

For d4, rb(Wd,Pd+1)=2d2.

Proof

Consider the following (2d3)-edge-coloring of Wd: all spokes are colored with d different colors and all rim edges are colored with d3 new colors arbitrarily. One can see that Wd does not have a rainbow Pd+1. Thus, rb(Wd,Pd+1)2d2. Note that the subgraph obtained from Wd 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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