The minimum size of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-rainbow connected graphs of given order

Bode, Jens-P.; Harborth, Heiko

doi:10.1016/j.disc.2012.07.023

Cited by 6 publications

(4 citation statements)

References 1 publication

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“…For n/2 < k, this theorem coincide with Theorem 1.1. As mentioned before, the case k = 3 has been already proved by Bode and Harborth [1], but our proof is different and shorter.…”

Section: Introductionmentioning

confidence: 61%

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A note on the minimum size ofk-rainbow-connected graphs

2014

Discrete Mathematics

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Abstract. An edge-coloured graph G is rainbow connected if there exists a rainbow path between any two vertices. A graph G is said to be k-rainbow connected if there exists an edge-colouring of G with at most k colours that is rainbow connected. For integers n and k, let t(n, k) denote the minimum number of edges in k-rainbow connected graphs of order n. In this note, we prove that t(n, k) = ⌈k(n − 2)/(k − 1)⌉ for all n, k ≥ 3.

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“…For n/2 < k, this theorem coincide with Theorem 1.1. As mentioned before, the case k = 3 has been already proved by Bode and Harborth [1], but our proof is different and shorter.…”

Section: Introductionmentioning

confidence: 61%

“…where the lower bound is due to Li et al [3] and the upper bound is due to a construction of Bode and Harborth [1]. When k = 3, Bode and Harborth [1] showed that t(n, 3) is actually equal to the upper bound for n ≥ 3. In this note, we show that the same statement holds for all n, k ≥ 3.…”

Section: Introductionmentioning

confidence: 98%

A note on the minimum size ofk-rainbow-connected graphs

2014

Discrete Mathematics

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“…[1,14,15,20]. Extremal problems have been studied in [3,21,30,33]. Vertex-rainbow connection number was introduced by Krivelevich and Yuster [19].…”

Section: Introductionmentioning

confidence: 99%

The vertex- rainbow connection number of some graph operations

Li¹,

Li²,

Ma³

2021

Discuss. Math. Graph Theory

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A path in an edge-colored (respectively vertex-colored) graph G is rainbow (respectively vertex-rainbow) if no two edges (respectively internal vertices) of the path are colored the same. An edge-colored (respectively vertex-colored) graph G is rainbow connected (respectively vertex-rainbow connected) if every two distinct vertices are connected by a rainbow (respectively vertex-rainbow) path. The rainbow connection number rc(G) (respectively vertex-rainbow connection number rvc(G)) of G is the smallest number of colors that are needed in order to make G rainbow connected (respectively vertex-rainbow connected). In this paper, we show that for a connected graph G known nontrivial inequality between the rainbow connection number and vertex-rainbow connection number. Moreover, the bounds reported are tight or tight up to additive constants.

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“…Problem 1 for k = 1 was first considered by Schiermeyer [19], when he determined the values of t 1 (n, r) for n 2 ≤ r ≤ n − 1, and the asymptotic answer for r = 2. Subsequently, Bode and Harborth [2], and Li et al [12] made some improvements, and Lo [17] determined the values of t 1 (n, r) for 3 ≤ r < n 2 . Thus Problem 1 is essentially completely solved for k = 1.…”

Section: Introductionmentioning

confidence: 99%

The size of graphs with restricted rainbow $2$-connection number

Fujita,

Liu,

Park

2019

Preprint

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Let k be a positive integer, and G be a k-connected graph. An edge-coloured path is rainbow if all of its edges have distinct colours. The rainbow k-connection number of G, denoted by rc k (G), is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by k internally vertex-disjoint rainbow paths. The function rc k (G) was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let t k (n, r) denote the minimum number of edges in a k-connected graph G on n vertices with rc k (G) ≤ r. Let s k (n, r) denote the maximum number of edges in a k-connected graph G on n vertices with rc k (G) ≥ r. The functions t 1 (n, r) and s 1 (n, r) have previously been studied by various authors. In this paper, we study the functions t 2 (n, r) and s 2 (n, r). We determine bounds for t 2 (n, r) which imply that t 2 (n, 2) = (1 + o(1))n log 2 n, and t 2 (n, r) is linear in n for r ≥ 3. We also provide some remarks about the function s 2 (n, r).

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The minimum size ofk-rainbow connected graphs of given order

Cited by 6 publications

References 1 publication

A note on the minimum size ofk-rainbow-connected graphs

A note on the minimum size ofk-rainbow-connected graphs

The vertex- rainbow connection number of some graph operations

The size of graphs with restricted rainbow $2$-connection number

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