- Email: [email protected]
On rainbow matchings in plane triangulations
Discrete Mathematics 342 (2019) 111624
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
On rainbow matchings in plane triangulations Stanislav Jendrol’ Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
article
info
a b s t r a c t
Article history: Received 10 October 2018 Received in revised form 25 July 2019 Accepted 28 July 2019 Available online xxxx
Let T n denote the class of all plane triangulations of order n. Denote by rb(Tn , kK2 ) the minimum number r of colors such that if kK2 ⊆ Tn ∈ Tn , then any edge-coloring of Tn with r colors contains a rainbow copy of kK2 . We show in this note that for any k ≥ 3,
Keywords: Rainbow matching Plane triangulation Edge-coloring
This improves the upper bound 2n + 6k − 16 recently received by Quin, Lan, and Shi. Note that, by Jendrol’ et al. (2014) for k ≥ 4,
rb(Tn , kK2 ) ≤ 2n + 4k − 13.
rb(Tn , kK2 ) ≥ 2n + 2k − 9.
© 2019 Published by Elsevier B.V.
1. Introduction All graphs in this note are undirected, finite and simple. We follow [3] for graph theoretical notations and terminology not defined here. Let G be a connected graph with vertex set V (G) and edge set E(G). For a set X ⊆ V (G) we define G[X ] to be the induced subgraph of G on the vertex set X . We use NG (x) to denote the set of vertices in G which are adjacent to x. We define degG (x) = |NG (x)| to be the degree of the vertex x. If G is edge-colored in a given way and a subgraph H of G contains no two edges of the same color, then H is called a rainbow subgraph of G or, in other words, a rainbow (copy of) H. Let f (G, H) denote the maximum number of colors in an edge-coloring of G with no rainbow copy of H. The number f (Kn , H) is called the anti-Ramsey number and has been introduced by Erdős, Simonovits and Sós in [5] (and denoted by f (n, H)). It is closely related to the rainbow number rb(G, H) representing the minimum number r of colors such that any edge-coloring of G with at least r colors contains a rainbow copy of H. Clearly, rb(G, H) = f (G, H) + 1. The rainbow number has been widely studied for different families of graphs G and H, see e.g. [1,2,4,6,8,11–18,21] or a survey [7]. The rainbow number for a matching kK2 of size k with respect to complete graphs Kn has been completely determined step by step in [4–6,21]. The rainbow number for matchings in 3-regular graphs is studied in [13], in some families of plane graphs in [15], and in hypergraphs in [19]. In this note we investigate the rainbow numbers when host graphs are plane triangulations. Let Tn be the class of all plane triangulations of order n. Denote by rb(Tn , H) the minimum number r of colors such that if H ⊆ Tn ∈ Tn , then any edge-coloring of Tn with r colors contains a rainbow copy of H. The study of rainbow numbers in plane triangulations was initiated by Horňák et al. [9] who investigated the rainbow numbers for cycles. Recently, Lan, Shi, and Song [16] improved some bounds for the rainbow numbers of cycles in plane triangulations and also obtained some new results for paths. Jendrol’, Schiermeyer, and Tu [10] initiated the study of rainbow numbers for matching in plane triangulations. They proved the following: E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111624 0012-365X/© 2019 Published by Elsevier B.V.
2
S. Jendrol’ / Discrete Mathematics 342 (2019) 111624
Theorem 1 ([10]). If k ≥ 5, then
(
2n + 2k − 9 ≤ rb(Tn , kK2 ) ≤ 2n + 2k − 7 + 2
)
2k − 2 3
.
They also determined the precise values of rb(Tn , kK2 ) for k = 2, k = 3, and for k = 4. Very recently, Qin, Lan, and Shi [20] have shown that rb(Tn , 5K2 ) = 2n + 1 for all n ≥ 11 and improved the upper bound from the above theorem. Namely, they proved: Theorem 2 ([20]). If k ≥ 5, then rb(Tn , kK2 ) ≤ 2n + 6k − 16. The aim of this note is to strengthen Theorem 2. We have the following: Theorem 3. If k ≥ 3, then rb(Tn , kK2 ) ≤ 2n + 4k − 13. 2. Proof of Theorem 3 We proceed by induction on k. The statement is true for k = 3 by the result of Jendrol’, Schiermeyer, and Tu in [10], who proved that rb(Tn , 3K2 ) = n + 1(≤ 2n + 12 − 13). Assume k ≥ 4. Let Tn ∈ Tn and let ϕ be an edge-coloring of Tn with 2n+4k−13 colors. As 2n+4k−13 > 2n+4(k−1)−13, the graph Tn contains a matching F = (k − 1)K2 by the induction hypothesis. Suppose, to the contrary, that Tn does not contain any rainbow copy of kK2 . Let G be a rainbow subgraph of Tn with 2n + 4k − 13 edges and having F as a subgraph. Clearly, G contains no rainbow copy of kK2 . Let M = {e1 , . . . , ek−1 } be a maximum matching in G and let V (M) be the set of end-vertices of the edges from M. Denote by xi this end vertex of ei which is adjacent to a vertex zi from the set B = V (G) \ V (M) and by yi the other end vertex of ei . Observe that yi is not adjacent to any vertex z ∈ (B \ {zi }); otherwise the edge set {zi xi , yi z } ∪ (M \ {ei }) forms a rainbow kK2 . A contradiction. If none of the end vertices of ei is adjacent to a vertex from B, we choose one of its vertex as xi and the other as yi . Observe, that there is no edge of G with both ends in B. Let X = {x1 , . . . , xk−1 } and Y = {y1 , . . . , yk−1 }. Then V (M) = X ∪ Y . Let us call vertices from the set X x-vertices. Observe that the subgraph H = G[B ∪ X ] \ E(G[X ]) is a plane bipartite graph. Observation 1. If for some i ∈ {1, . . . , k − 1} there exist an edge ei = xi yi in M and edges zi xi and zi yi in G, then degH (xi ) = 1. Proof. Otherwise there would exist a vertex w in B, w ̸ = zi , and an edge w xi . But then the matching ({w xi , zi yi }) ∪ (M \{ei }) would have k edges, a contradiction. □ Let a denote the number of x-vertices with degH (xi ) = 0, b the number of x-vertices with degH (xi ) = 1, and let c be the number of x-vertices with degH (xj ) ≥ 2. Denote by d the number of edges ei from M having the property described in Observation 1 (Note that in fact, d is equal to the number of zi yi edges in G). Then
• k − 1 = a + b + c, • d ≤ b, • |E(H)| ≤ 2(n − 2k + 2 + c) − 4 + b = 2n − 4k + 2c + b (Here we have used the fact that for any bipartite graph K = (V (K ), E(K )) there is |E(K )| ≤ 2|V (K )| − 4), and • |E(G[X ∪ Y ])| ≤ 3(2k − 2) − 6 = 6k − 12. The above considerations and observations lead to 2n + 4k − 13 = |E(G)| =
= |E(H)| + d + |E(G[X ∪ Y ])| ≤ 2n + 2k + 2c + b + d − 12 ≤ 2n + 2k + 2(b + c) − 12 ≤ 2n + 4k − 2a − 14, a contradiction. This finishes the proof of our Theorem 3. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the Slovak Research and Development Agency under the Contract No. APVV-15-0116 and by the Slovak VEGA Grant 1/0368/16.
S. Jendrol’ / Discrete Mathematics 342 (2019) 111624
3
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
N. Alon, On a conjecture of Erdős, Simonovits and Sós concerning anti-Ramsey theorems, J. Graph Theory 7 (1983) 91–94. M. Axenovich, T. Jiang, A. Kündgen, Bipartite anti-Ramsey numbers for cycles, J. Graph Theory 47 (2004) 9–28. J.A. Bondy, U.S.R. Murty, Graph Theory, in: GTM, vol. 244, Springer, 2008. H. Chen, X. Li, J. Tu, Complete solution for the rainbow numbers of matchings, Discrete Math. 309 (2009) 3370–3380. P. Erdős, M. Simonovits, V.T. Sós, Anti-Ramsey Theorems, in: Colloq. Math. Soc. János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 633–643. S. Fujita, A. Kaneko, I. Schiermeyer, K. Suzuki, A rainbow k-matching in the complete graph with r colors, Electron. J. Combin. 16 (2009) R51. S. Fujita, C. Magnant, K. Ozeki, Rainbow generalization of Ramsey theory: a survey, Graphs Combin. 26 (2010) 1–30. R. Haas, M. Young, The anti-Ramsey number of perfect matching, Discrete Math. 312 (2012) 933–937. M. Horňák, S. Jendrol’, I. Schiermeyer, R. Soták, Rainbow numbers for cycles in plane triangulations, J. Graph Theory 78 (2015) 248–257. S. Jendrol’, I. Schiermeyer, J. Tu, Rainbow matching in plane triangulations, Discrete Math. 331 (2014) 158–164. T. Jiang, Anti-Ramsey numbers of subdivided graphs, J. Combin. Theory Ser. B. 85 (2002) 361–366. T. Jiang, D.B. West, On the Erdős-Simonovits-Sós conjecture about anti-Ramsey number of a cycle, Combin. Probab. Comput. 12 (2003) 585–598. Z. Jin, Anti-Ramsey numbers for matchings in 3-regular bipartite graphs, Appl. Math. Comput. 292 (2009) 114–119. Z. Jin, X. Li, Anti-Ramsey numbers for graphs with independent cycles, Electron. J. Combin. 16 (2009) R85. Z. Jin, K. Ye, Rainbow number of matchings in planar graphs, Discrete Math. 341 (2018) 2846–2858. Y. Lan, Y. Shi, Z.-X. Song, Planar anti-Ramsey numbers for paths and cycles, arXiv:1709.00970v1. X. Li, J. Tu, Z. Jin, Bipartite rainbow numbers of matchings, Discrete Math. 309 (2009) 2575–2578. J.J. Montellano-Ballesteros, V. Neumann-Lara, An anti-Ramsey theorem, Combinatorica 22 (2002) 445–449. L. Özkahya, M. Young, Anti-Ramsey number of matchings in hypergraphs, Discrete Math. 313 (2013) 2359–2364. Z. Qin, Y. Lan, Y. Shi, Improved bounds for rainbow numbers of matchings in plane triangulations, Discrete Math. 343 (2019) 221–225. I. Schiermeyer, Ramsey numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157–162.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
On rainbow matchings in plane triangulations Stanislav Jendrol’ Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
article
info
a b s t r a c t
Article history: Received 10 October 2018 Received in revised form 25 July 2019 Accepted 28 July 2019 Available online xxxx
Let T n denote the class of all plane triangulations of order n. Denote by rb(Tn , kK2 ) the minimum number r of colors such that if kK2 ⊆ Tn ∈ Tn , then any edge-coloring of Tn with r colors contains a rainbow copy of kK2 . We show in this note that for any k ≥ 3,
Keywords: Rainbow matching Plane triangulation Edge-coloring
This improves the upper bound 2n + 6k − 16 recently received by Quin, Lan, and Shi. Note that, by Jendrol’ et al. (2014) for k ≥ 4,
rb(Tn , kK2 ) ≤ 2n + 4k − 13.
rb(Tn , kK2 ) ≥ 2n + 2k − 9.
© 2019 Published by Elsevier B.V.
1. Introduction All graphs in this note are undirected, finite and simple. We follow [3] for graph theoretical notations and terminology not defined here. Let G be a connected graph with vertex set V (G) and edge set E(G). For a set X ⊆ V (G) we define G[X ] to be the induced subgraph of G on the vertex set X . We use NG (x) to denote the set of vertices in G which are adjacent to x. We define degG (x) = |NG (x)| to be the degree of the vertex x. If G is edge-colored in a given way and a subgraph H of G contains no two edges of the same color, then H is called a rainbow subgraph of G or, in other words, a rainbow (copy of) H. Let f (G, H) denote the maximum number of colors in an edge-coloring of G with no rainbow copy of H. The number f (Kn , H) is called the anti-Ramsey number and has been introduced by Erdős, Simonovits and Sós in [5] (and denoted by f (n, H)). It is closely related to the rainbow number rb(G, H) representing the minimum number r of colors such that any edge-coloring of G with at least r colors contains a rainbow copy of H. Clearly, rb(G, H) = f (G, H) + 1. The rainbow number has been widely studied for different families of graphs G and H, see e.g. [1,2,4,6,8,11–18,21] or a survey [7]. The rainbow number for a matching kK2 of size k with respect to complete graphs Kn has been completely determined step by step in [4–6,21]. The rainbow number for matchings in 3-regular graphs is studied in [13], in some families of plane graphs in [15], and in hypergraphs in [19]. In this note we investigate the rainbow numbers when host graphs are plane triangulations. Let Tn be the class of all plane triangulations of order n. Denote by rb(Tn , H) the minimum number r of colors such that if H ⊆ Tn ∈ Tn , then any edge-coloring of Tn with r colors contains a rainbow copy of H. The study of rainbow numbers in plane triangulations was initiated by Horňák et al. [9] who investigated the rainbow numbers for cycles. Recently, Lan, Shi, and Song [16] improved some bounds for the rainbow numbers of cycles in plane triangulations and also obtained some new results for paths. Jendrol’, Schiermeyer, and Tu [10] initiated the study of rainbow numbers for matching in plane triangulations. They proved the following: E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111624 0012-365X/© 2019 Published by Elsevier B.V.
2
S. Jendrol’ / Discrete Mathematics 342 (2019) 111624
Theorem 1 ([10]). If k ≥ 5, then
(
2n + 2k − 9 ≤ rb(Tn , kK2 ) ≤ 2n + 2k − 7 + 2
)
2k − 2 3
.
They also determined the precise values of rb(Tn , kK2 ) for k = 2, k = 3, and for k = 4. Very recently, Qin, Lan, and Shi [20] have shown that rb(Tn , 5K2 ) = 2n + 1 for all n ≥ 11 and improved the upper bound from the above theorem. Namely, they proved: Theorem 2 ([20]). If k ≥ 5, then rb(Tn , kK2 ) ≤ 2n + 6k − 16. The aim of this note is to strengthen Theorem 2. We have the following: Theorem 3. If k ≥ 3, then rb(Tn , kK2 ) ≤ 2n + 4k − 13. 2. Proof of Theorem 3 We proceed by induction on k. The statement is true for k = 3 by the result of Jendrol’, Schiermeyer, and Tu in [10], who proved that rb(Tn , 3K2 ) = n + 1(≤ 2n + 12 − 13). Assume k ≥ 4. Let Tn ∈ Tn and let ϕ be an edge-coloring of Tn with 2n+4k−13 colors. As 2n+4k−13 > 2n+4(k−1)−13, the graph Tn contains a matching F = (k − 1)K2 by the induction hypothesis. Suppose, to the contrary, that Tn does not contain any rainbow copy of kK2 . Let G be a rainbow subgraph of Tn with 2n + 4k − 13 edges and having F as a subgraph. Clearly, G contains no rainbow copy of kK2 . Let M = {e1 , . . . , ek−1 } be a maximum matching in G and let V (M) be the set of end-vertices of the edges from M. Denote by xi this end vertex of ei which is adjacent to a vertex zi from the set B = V (G) \ V (M) and by yi the other end vertex of ei . Observe that yi is not adjacent to any vertex z ∈ (B \ {zi }); otherwise the edge set {zi xi , yi z } ∪ (M \ {ei }) forms a rainbow kK2 . A contradiction. If none of the end vertices of ei is adjacent to a vertex from B, we choose one of its vertex as xi and the other as yi . Observe, that there is no edge of G with both ends in B. Let X = {x1 , . . . , xk−1 } and Y = {y1 , . . . , yk−1 }. Then V (M) = X ∪ Y . Let us call vertices from the set X x-vertices. Observe that the subgraph H = G[B ∪ X ] \ E(G[X ]) is a plane bipartite graph. Observation 1. If for some i ∈ {1, . . . , k − 1} there exist an edge ei = xi yi in M and edges zi xi and zi yi in G, then degH (xi ) = 1. Proof. Otherwise there would exist a vertex w in B, w ̸ = zi , and an edge w xi . But then the matching ({w xi , zi yi }) ∪ (M \{ei }) would have k edges, a contradiction. □ Let a denote the number of x-vertices with degH (xi ) = 0, b the number of x-vertices with degH (xi ) = 1, and let c be the number of x-vertices with degH (xj ) ≥ 2. Denote by d the number of edges ei from M having the property described in Observation 1 (Note that in fact, d is equal to the number of zi yi edges in G). Then
• k − 1 = a + b + c, • d ≤ b, • |E(H)| ≤ 2(n − 2k + 2 + c) − 4 + b = 2n − 4k + 2c + b (Here we have used the fact that for any bipartite graph K = (V (K ), E(K )) there is |E(K )| ≤ 2|V (K )| − 4), and • |E(G[X ∪ Y ])| ≤ 3(2k − 2) − 6 = 6k − 12. The above considerations and observations lead to 2n + 4k − 13 = |E(G)| =
= |E(H)| + d + |E(G[X ∪ Y ])| ≤ 2n + 2k + 2c + b + d − 12 ≤ 2n + 2k + 2(b + c) − 12 ≤ 2n + 4k − 2a − 14, a contradiction. This finishes the proof of our Theorem 3. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the Slovak Research and Development Agency under the Contract No. APVV-15-0116 and by the Slovak VEGA Grant 1/0368/16.
S. Jendrol’ / Discrete Mathematics 342 (2019) 111624
3
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
N. Alon, On a conjecture of Erdős, Simonovits and Sós concerning anti-Ramsey theorems, J. Graph Theory 7 (1983) 91–94. M. Axenovich, T. Jiang, A. Kündgen, Bipartite anti-Ramsey numbers for cycles, J. Graph Theory 47 (2004) 9–28. J.A. Bondy, U.S.R. Murty, Graph Theory, in: GTM, vol. 244, Springer, 2008. H. Chen, X. Li, J. Tu, Complete solution for the rainbow numbers of matchings, Discrete Math. 309 (2009) 3370–3380. P. Erdős, M. Simonovits, V.T. Sós, Anti-Ramsey Theorems, in: Colloq. Math. Soc. János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 633–643. S. Fujita, A. Kaneko, I. Schiermeyer, K. Suzuki, A rainbow k-matching in the complete graph with r colors, Electron. J. Combin. 16 (2009) R51. S. Fujita, C. Magnant, K. Ozeki, Rainbow generalization of Ramsey theory: a survey, Graphs Combin. 26 (2010) 1–30. R. Haas, M. Young, The anti-Ramsey number of perfect matching, Discrete Math. 312 (2012) 933–937. M. Horňák, S. Jendrol’, I. Schiermeyer, R. Soták, Rainbow numbers for cycles in plane triangulations, J. Graph Theory 78 (2015) 248–257. S. Jendrol’, I. Schiermeyer, J. Tu, Rainbow matching in plane triangulations, Discrete Math. 331 (2014) 158–164. T. Jiang, Anti-Ramsey numbers of subdivided graphs, J. Combin. Theory Ser. B. 85 (2002) 361–366. T. Jiang, D.B. West, On the Erdős-Simonovits-Sós conjecture about anti-Ramsey number of a cycle, Combin. Probab. Comput. 12 (2003) 585–598. Z. Jin, Anti-Ramsey numbers for matchings in 3-regular bipartite graphs, Appl. Math. Comput. 292 (2009) 114–119. Z. Jin, X. Li, Anti-Ramsey numbers for graphs with independent cycles, Electron. J. Combin. 16 (2009) R85. Z. Jin, K. Ye, Rainbow number of matchings in planar graphs, Discrete Math. 341 (2018) 2846–2858. Y. Lan, Y. Shi, Z.-X. Song, Planar anti-Ramsey numbers for paths and cycles, arXiv:1709.00970v1. X. Li, J. Tu, Z. Jin, Bipartite rainbow numbers of matchings, Discrete Math. 309 (2009) 2575–2578. J.J. Montellano-Ballesteros, V. Neumann-Lara, An anti-Ramsey theorem, Combinatorica 22 (2002) 445–449. L. Özkahya, M. Young, Anti-Ramsey number of matchings in hypergraphs, Discrete Math. 313 (2013) 2359–2364. Z. Qin, Y. Lan, Y. Shi, Improved bounds for rainbow numbers of matchings in plane triangulations, Discrete Math. 343 (2019) 221–225. I. Schiermeyer, Ramsey numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157–162.