Planar anti-Ramsey numbers of matchings

Planar anti-Ramsey numbers of matchings

Gang Chen, Yongxin Lan and Zi-Xia Song
School of mathematics and statistics
Ningxia University, China
Center for Combinatorics and LPMC
Nankai University, Tianjin, 300071, China
Department of Mathematics
University of Central Florida
Orlando, FL 32816, USA
Abstract

Given a positive integer and a planar graph , let be the family of all plane triangulations on vertices such that contains a subgraph isomorphic to . The planar anti-Ramsey number of , denoted , is the maximum number of colors in an edge-coloring of a plane triangulation such that contains no rainbow copy of . In this paper we study planar anti-Ramsey numbers of matchings. For all , let denote a matching of size . We prove that for all and , , which significantly improves the existing lower and upper bounds for . It seems that for each , the lower bound we obtained is the exact value of for sufficiently large . This is indeed the case for . We prove that for all .

1 Introduction

All graphs considered in this paper are finite and simple. For a graph we use and to the number of vertices and number of edges of , respectively. For a vertex , we will use to denote the set of vertices in which are adjacent to . We define . For any , the subgraph of induced by , denoted , is the graph with vertex set and edge set . We denote the subgraph of induced on . If , we simply write . For disjoint subsets of , we use to denote the number of edges in with one end in and the other in . Since every planar bipartite graph on vertices has at most edges, we will frequently use the fact that when is planar and . Given two isomorphic graphs and , we may (with a slight but common abuse of notation) write . For any positive integer , let . We use the convention that “” means that is defined to be the right-hand side of the relation.

Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós [4] in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a planar graph and a positive integer , let be the family of all plane triangulations on vertices such that contains a subgraph isomorphic to . The planar anti-Ramsey number of , denoted , is the maximum number of colors in an edge-coloring of a plane triangulation such that contains no rainbow copy of . Analogous to the relation between anti-Ramsey numbers and Turán numbers proved in [4], planar anti-Ramsey numbers are closely related to planar Turán numbers [13], where the planar Turán number of , denoted , is the maximum number of edges of a planar graph on vertices that contains no subgraph isomorphic to .

Proposition 1.1 ([13])

Given a planar graph and a positive integer ,

 1+exP(n,H)≤arP(n,H)≤exP(n,H),

where .

Dowden [3] began the study of planar Turán numbers (under the name of “extremal” planar graphs). Results on planar Turán numbers of paths and cycles can be found in [3, 12]. The study of planar anti-Ramsey numbers was initiated by Horňák, Jendrol, Schiermeyer and Soták [6] (under the name of rainbow numbers). Results on planar anti-Ramsey numbers of paths and cycles can be found in [6, 13]. Colorings of plane graphs that avoid rainbow faces have also been studied, see, e.g., [5, 7, 17, 18]. Various results on anti-Ramsey numbers can be found in: [1, 2, 9, 10, 11, 14, 16] to name a few.

Finding exact values of is far from trivial. As observed in [6], an induction argument in general cannot be applied to compute because deleting a vertex from a plane triangulation may result in a graph that is no longer a plane triangulation. In this paper, we study planar anti-Ramsey numbers of matchings. For all , let denote a matching of size . In [8], the exact value of when was determined, and lower and upper bounds for were also established for all and . Recently, the exact value of was determined in [15] and an improved upper bound for was also obtained in [15]. We summarize the results in [8, 15] below.

Theorem 1.2 ([8])

Let and be positive integers. Then

1. for all , .

2. for all , .

3. for all and , .

Theorem 1.3 ([15])

Let and be positive integers. Then

1. for all , .

2. for all and , .

In this paper, we further improve the existing lower and upper bounds for .

Theorem 1.4

For all and , .

Theorem 1.4 significantly improves the lower bound in Theorem 1.2(c) and the new upper bound in Theorem 1.3(b). We believe that for each , the lower bound we obtained in Theorem 1.4 is the exact value of for sufficiently large . This is indeed the case for .

Theorem 1.5

For all , .

It seems that the method we developed in the proof of Theorem 1.5 can be applied to close the gap in Theorem 1.4. We prove Theorem 1.4 in Section 2 and Theorem 1.5 in Section 3.

2 Proof of Theorem 1.4

We are ready to prove Theorem 1.4. Let be given as in the statement. We first prove that . Let be a path with vertices in order. Let be the plane triangulation obtained from by adding two adjacent vertices and joining each of and to all the vertices on with the outer face of having vertices on its boundary. Then and is hamiltonian. Let be the plane triangulation obtained from by adding a new vertex to each face of and then joining it to all vertices on the boundary of . Then is a plane triangulation on vertices. Let be the new vertex added to the outer-face of . Let be the plane triangulation on vertices obtained from by adding vertices, say , to the face of containing , such that , and for all , is adjacent to in . The construction of when and is depicted in Figure 1. Clearly, . Let be an edge-coloring of by first coloring all the edges by color and then all the remaining edges of by distinct colors other than . It can be easily checked that has no rainbow under the coloring and the total number of colors used by is . This proves that , as desired.

It remains to prove that . Suppose for some and . Then there exists a such that has no rainbow copy of under some onto mapping , where . We choose such a with minimum. Let be a rainbow spanning subgraph of with edges. Then does not contain because has no rainbow copy of . By minimality of and Theorem 1.3(a) (when ), contains a copy of . Let be a matching of size in , and let . Let . For each , we may assume that . Since is the largest matching in , we see that has no -augmenting path. It follows that has no edges, and for each , either or with . We may further assume that are such that for all , and for all , where . Then . Let . Then when , and when because is a planar bipartite graph on vertices. Since is planar and , we have . Thus, , contrary to . This completes the proof of Theorem 1.4.

Remark. For , the condition “” in the statement of Theorem 1.4 can be replaced by “”.

3 Proof of Theorem 1.5

We need to introduce more notation that shall be used in this section only. For , let be the set of all plane triangulations on vertices, and let be the set of all planar graphs with vertices and edges. Clearly, every graph in is isomorphic to a plane triangulation on vertices with one edge removed. By abusing notation, let and . It is known that every plane triangulation on vertices is -connected. It is also known that every plane triangulation on vertices has a Hamilton cycle and every plane triangulation on vertices does not necessarily have a Hamilton cycle111The third author would like to thank Jason Bentley, a Ph.D. student at the University of Central Florida, for his help in carefully verifying these facts with her.. We summarize these facts as follows.

Observation 3.1

Let be a planar triangulation on vertices. Then

1. is -connected.

2. for every , has a Hamilton cycle.

3. for every , does not necessarily have a Hamilton cycle.

Let denote the number of odd components in a graph . We shall make use of the following theorem in the proof of Theorem 1.5.

Theorem 3.2 (Berge-Tutte Formula)

Let be a graph on vertices and let be the size of a maximum matching of . Then there exists an with such that

 2d=n−o(G∖S)+|S|.

Moreover, each odd component of is factor-critical.

Proof of Theorem 1.5: Let be an integer. By Theorem 1.4, . We next show that . Suppose . Then there exists a such that has no rainbow under some onto mapping , where . Let be a rainbow spanning subgraph of with edges. By Theorem 1.3(a), has a copy of . Clearly, has no copy of because has no rainbow copy of under . By Theorem 3.2, there exists an with such that . Let be all the odd components of . We may assume that . Let . Then . It follows that . Let when and for all . We may further assume that . Let and . Then and when . We next prove several claims.

Claim 1.  If has two edge-disjoint matchings of size , say and , then has no edges.

Proof. Suppose has an edge . We may assume that for all . But then is a rainbow in under the coloring , a contradiction.

Claim 2.  If , then .

Proof. Suppose . Then has a Hamilton cycle by Observation 3.1(b), and thus has two edge-disjoint matchings of size . By Claim 1, has no edges. But then

 e(T) =e(T[W∪S∪{u1,…,u10−w−s}])+eT({u11−w−s,…,ur},W∪S∪{u1,…,u10−w−s}) ≤e(T10)+(2n−4)=24+(2n−4),

which implies that because , contrary to .

Claim 3..

Proof. Suppose . Then and so . It follows that

 2n+4≤e(G) =e(G[W∪S])+eG(U,S)≤(3(w+s)−6)+(2(n−w)−4) =2n+w+3s−10=2n+(10−2s)+3s−10=2n+s,

which implies that . If , then . But then

 2n+4≤e(G)=e(G[S])+e(G[W])+eG(S,U∪W)≤e(T4)+1+2n−4=2n+3,

which is impossible. Thus , and so and . By Claim 2, . Since , we see that . Thus or . Then , else , a contradiction. Suppose . Then and . It follows that , else . But then , contrary to . This proves that . We may assume that . Then , and , which implies that . If , then , a contradiction. Thus . Furthermore, , else , contrary to . Since , we see that . Then , else , a contradiction. Since does not contain as a subgraph, we may assume that . Then because . Note that . By Observation 3.1(b), has a hamiltonian path with as an end. Since , we see that has two edge-disjoint matchings of size . By Claim 1, has no edges. But then

 3n−6=e(T) =e(T[S∪{u1,…,u6}])+eT({u7,…,ur},S∪{u1,…,u6}) ≤e(T11)+(2n−4)=27+(2n−4),

which implies that , contrary to .

By Claim 3, and . Then , which implies that , with only when .

Claim 4..

Proof. Suppose . By Claim 3, . Thus . It follows that

 2n+4≤e(G) =e(G[W∪S])+eG(U,S)≤(3(w+s)−6)+(2(n−w)−4) =2n+w+3s−10=2n+(11−2s)+3s−10=2n+s+1,

which implies that . If , then and so , else , a contradiction. Since does not contain as a subgraph, we see that . By Claim 2, . Thus . Note that . But then

 2n+4≤e(G)=e(G[W∪S])+eG(U,S)≤(20−dG(u2))+(2n−17+dG(u2))=2n+3,

which is impossible. Thus . Then . It follows that and because is factor-critical. Then , else and so , a contradiction. Since , we see that . Thus or . Suppose . Then for all . Since , we see that . But then , contrary to . This proves that . Then , else , a contradiction. Since , we see that . Then because . Since and , we may assume that for some with . Note that does not contain as a subgraph. We may further assume that . Then . By Observation 3.1(b), has a hamiltonian path with as an end. Since , we see that has two edge-disjoint matchings of size . By Claim 1, has no edges. But then

 3n−6=e(T) =e(T[W∪S∪{u1,u2,u3,u4}])+eT({u5,…,ur},W∪S∪{u1,u2,u3,u4}) ≤e(T11)+(2n−4)=27+(2n−4),

which implies that , contrary to .

By Claim 4, and . Then , which implies that , with only when and . Suppose . Then . By Claim 2, . Since does not contain as a subgraph, we see that . Then and , else either

 e(G)=e(H)+eG({u2,…,ur},S)≤23+2(n−10)=2n+3, or
 e(G)=e(H)+eG({u2,…,ur},S)≤22+3+2(n−11)=2n+3,

contrary to in both cases. Let and . Since , we may assume that . It follows that has two edge-disjoint matchings of size , namely, and . By Claim 1, has no edges. But then

 3n−6=e(T) =e(T[W∪S∪{u1,u2}])+eT({u3,…,ur},W∪S∪{u1,u2}) ≤e(T11)+(2n−4)=27+(2n−4),

which implies that , contrary to . Thus . Then for all and is a vertex-cut of because . Then and . But then

 2n+4≤e(G) =eG(U,S)+e(G[S∪W]) ≤rs+3(n−r)−8 =3n−(3−s)r−8 ≤3n−(3−s)(n+2s−15)−8,

which is impossible because and . This completes the proof of Theorem 1.5.

Remark. In the proof of Theorem 1.5, Claim 1 is applied to two vertex-disjoint matchings, instead of edge-disjoint matchings. It seems that the method we developed in the proof of Theorem 1.5 can be used to close the gap in Theorem 1.4.

Acknowledgments. The authors would like to thank Jingmei Zhang for helpful comments.
Gang Chen would like to thank the University of Central Florida for hosting his visit. His research is partially supported by NSFC under the grant number 71561022 and Overseas Training Program for Faculty at Ningxia University.

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