Improved bounds for rainbow numbers of matchings in plane triangulations

Abstract

Given two graphs G and H, the rainbow number rb(G, H) for H with respect to G is defined as the minimum number k such that any k-edge-coloring of G contains a rainbow H, i.e., a copy of H, all of whose edges have different colors. Denote by kK 2 a matching of size k and T n the class of all plane triangulations of order n, respectively. In [S. Jendrol ′ , I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158-164], the authors determined the exact values of rb(T n , kK 2 ) for 2 ≤ k ≤ 4 and proved that 2n + 2k − 9 ≤ rb(T n , kK 2 ) ≤ 2n + 2k − 7 + 2 2k−2 3 for k ≥ 5. In this paper, we improve the upper bounds and prove that rb(T n , kK 2 ) ≤ 2n + 6k − 16 for n ≥ 2k and k ≥ 5. Especially, we show that rb(T n , 5K 2 ) = 2n + 1 for n ≥ 11.

Exact rainbow numbers for matchings in plane triangulations

“…Recently, the exact value of rb(T n , M 6 ) and an improved lower and upper bounds for rb(T n , M t ) were also obtained by Chen, Lan and Song in [4]. We summarize the known results in [4,14,24] as follows.…”

“…For all t ≥ 1, let M t denote a matching of size t. In [14], the exact values of rb(T n , M t ) when t ≤ 4 were determined, and lower and upper bounds for rb(T n , M t ) were also established for all t ≥ 5 and n ≥ 2t. In [24], the exact value of rb(T n , M 5 ) was determined and an improved upper bound for rb(T n , M t ) was obtained. Recently, the exact value of rb(T n , M 6 ) and an improved lower and upper bounds for rb(T n , M t ) were also obtained by Chen, Lan and Song in [4].…”

Planar anti-Ramsey numbers of matchings

“…For all t ≥ 1, let M t denote a matching of size t. In [8], the exact value of ar P (n, M t ) when t ≤ 4 was determined, and lower and upper bounds for ar P (n, M t ) were also established for all t ≥ 5 and n ≥ 2t. Recently, the exact value of ar P (n, M 5 ) was determined in [15] and an improved upper bound for ar P (n, M t ) was also obtained in [15]. We summarize the results in [8,15] below.…”

“…Recently, the exact value of ar P (n, M 5 ) was determined in [15] and an improved upper bound for ar P (n, M t ) was also obtained in [15]. We summarize the results in [8,15] below. In this paper, we further improve the existing lower and upper bounds for ar P (n, M t ).…”

The Outer-Planar Anti-Ramsey Number of Matchings

“…In 2014, Jendrol' et al [18] investigated the planar anti-Ramsey number of kK 2 , in which the upper and lower bounds of ar(T n , kK 2 ) for all k ≥ 5 and n ≥ 2k were established, and the exact values of ar(T n , kK 2 ) for 2 ≤ k ≤ 4 and n ≥ 2k were determined. Qin et al [19] improved the upper bound of ar(T n , kK 2 ) in [18] and determined the exact value of ar(T n , 5K 2 ) for all n ≥ 11. Later, Chen et al [20] improved the upper and lower bounds of ar(T n , kK 2 ) for k ≥ 6 and n ≥ 3k − 6 existing in [18,19], and determined the exact value of ar(T n , 6K 2 ) for all n ≥ 30.…”