An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Xueliang Li, Yuefang Sun
2017 Theory and Applications of Graphs  
The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008 . Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an
more » ... ion, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow k-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study. Li and Sun: Rainbow connections of graphs -a dynamic survey Li and Sun: Rainbow connections of graphs -a dynamic survey Published by Digital Commons@Georgia Southern, 2017, Last updated 2017 σ 2 . Similarly, they [42] further got the following result for a general k: If G is a connected graph of order with k independent vertices, then rc(G) ≤ 3kn σ k +k + 6k − 3. With respect to the the relation between rc(G) and the connectivity κ(G), mentioned in [133] , Broersma asked a question at the IWOCA workshop: Problem 2.2. [133] What happens with the value rc(G) for graphs with higher connectivity? Li and Liu got a best possible upper bound in [95, 97] for 2-connected graphs: Let G be a 2-connected graph of order n (n ≥ 3), then rc(G) ≤ ⌈ n 2 ⌉ and the upper bound is tight for n ≥ 4. In [47] , Ekstein et al. rediscovered this result. One could think of generalizing the above result to the case of higher connectivity. Li and Liu [95, 97] raised the following stronger conjecture that for every κ ≥ 1, if G is a κ-connected graph of order n, then rc(G) ≤ ⌈ n κ ⌉. Unfortunately, Ekstein et al. in [47] found examples showing that for every κ there are κ-connected graphs G of order n with rc(G) ≥ n−2 κ + 1, which is slightly bigger than ⌈ n κ ⌉ when κ (≥ 3) divides n.
doi:10.20429/tag.2017.000103 fatcat:76awzxcw6ra3ldnibx5xtvtmmu