Graphs with rainbow connection number two

Kemnitz, Arnfried; Schiermeyer, Ingo

doi:10.7151/dmgt.1547

Cited by 45 publications

(21 citation statements)

References 7 publications

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“…Those type of sufficient conditions are known as Erdős-Gallai type results. Research on the following Erdős-Gallai type problem has been started in [34].…”

Section: Given a Graph G And A Setmentioning

confidence: 99%

Rainbow Connections of Graphs: A Survey

Shi

Sun

2012

Graphs and Combinatorics

241

147

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The concept of rainbow connection was introduced by . It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

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“…Those type of sufficient conditions are known as Erdős-Gallai type results. Research on the following Erdős-Gallai type problem has been started in [34].…”

Section: Given a Graph G And A Setmentioning

confidence: 99%

Rainbow Connections of Graphs: A Survey

Shi

Sun

2012

Graphs and Combinatorics

241

147

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“…The analogous problem for rainbow connections was introduced in [12] and results on that problem appeared in [11,12,13,14,16].…”

Section: Introductionmentioning

confidence: 99%

Minimum degree and size conditions for the proper connection number of graphs

Guan

Xue

Cheng

et al. 2019

Applied Mathematics and Computation

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An edge-coloured graph G is called properly connected if every two vertices are connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number tobe at most k in terms of the size of graphs. In this note, our main result is the following, by adding a minimum degree condition: Let G be a connected graph of order n, k ≥ 3. IfFurthermore, if k = 2 and δ = 2, pc(G) ≤ 2, except G ∈ {G 1 , G n } (n ≥ 8), where G 1 = K 1 ∨ 3K 2 and G n is obtained by taking a complete graph K n−5 and K 1 ∨ (2K 2 ) with an arbitrary vertex of K n−5 and a vertex with d(v) = 4 in K 1 ∨ (2K 2 ) being joined. If k = 2, δ ≥ 3, we conjecture pc(G) ≤ 2, where m takes the value 1 if δ = 3 and 0 if δ ≥ 4 in the assumption.

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“…Chartrand et al computed the precise rainbow connection number of several graph classes including complete multipartite graphs ( [5]). The rainbow connection number has been studied for further graph classes in [2,9,14,15] and for graphs with fixed minimum degree in [2,10,18]. There are also some results on the aspect of extremal graph theory, such as [19].…”

Section: Introductionmentioning

confidence: 99%

Note on the Hardness of Rainbow Connections for Planar and Line Graphs

Huang

Shi

2014

Bull. Malays. Math. Sci. Soc.

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An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rc(G) is an NP-Hard problem, as well as that even deciding whether a graph has rc(G) = 2 is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph is a planar bipartite graph. We also give upper bounds of the rainbow connection number of outerplanar graphs with small diameters. A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. It is known that deciding whether a given vertex-colored graph is rainbow vertex-connected is NP-Complete. We will prove that it is still NP-Complete even when the vertex-colored graph is a line graph.

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Graphs with rainbow connection number two

Cited by 45 publications

References 7 publications

Rainbow Connections of Graphs: A Survey

Rainbow Connections of Graphs: A Survey

Minimum degree and size conditions for the proper connection number of graphs

Note on the Hardness of Rainbow Connections for Planar and Line Graphs

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