Rainbow Connections of Graphs: A Survey
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This survey attempts to bring together most of the results and papers that dealt with the concept of rainbow connection, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity.Abstract:
The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is 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 and questions.read more
Citations
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The generalized 3-connectivity of star graphs and bubble-sort graphs
Shasha Li,Jianhua Tu,Chenyan Yu +2 more
TL;DR: Two classes of Cayley graphs, the star graphs Sn and the bubble-sort graphs Bn, are restricted to and the generalized 3-connectivity of Sn and Bn is investigated, showing that ?
Journal ArticleDOI
Total rainbow k-connection in graphs
TL;DR: The total rainbow k -connection number of G , denoted by t r c k ( G ) , is the minimum number of colours required to colour the edges and vertices of G, so that any two vertice of G are connected by k internally vertex-disjoint total-rainbow paths.
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The strong rainbow vertex-connection of graphs
TL;DR: The strong rainbow vertex-connection number of a vertex-colored graph is the smallest number of colors that is needed in order to make the graph strongly rainbow vertex connected as discussed by the authors.
Journal ArticleDOI
Rainbow Connection of Random Regular Graphs
TL;DR: It is proved that with probability tending to one as $n$ goes to infinity the rainbow connection of G satisfies $rc(G)=O(\log n)$, which is best possible up to a hidden constant.
Journal ArticleDOI
Proper connection number of random graphs
Ran Gu,Xueliang Li,Zhongmei Qin +2 more
TL;DR: It is shown that almost all graphs have the proper connection number 2, defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of G is connected by at least one proper path in G.
References
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Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Book
Graph Theory
J. A. Bondy,U.S.R Murty +1 more
TL;DR: This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science.
Book
An Introduction to the Theory of Groups
TL;DR: The first six chapters provide ample material for a first course: beginning with the basic properties of groups and homomorphisms, topics covered include Lagrange's theorem, the Noether isomorphism theorems, symmetric groups, G-sets, the Sylow theorem, finite Abelian groups, the Krull-Schmidt theorem, solvable and nilpotent groups, and the Jordan-Holder theorem.
Book ChapterDOI
Congruent Graphs and the Connectivity of Graphs
TL;DR: In this paper, the authors give conditions that two graphs be congruent and some theorems on the connectivity of graphs, and conclude with some applications to dual graphs, which can also be proved by topological methods.