12 research outputs found
Rainbow Connection of Random Regular Graphs
An edge colored graph is rainbow edge connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection of
a connected graph , denoted by , is the smallest number of colors
that are needed in order to make rainbow connected.
In this work we study the rainbow connection of the random -regular graph
of order , where is a constant. We prove that with
probability tending to one as goes to infinity the rainbow connection of
satisfies , which is best possible up to a hidden
constant
Colorful Hamilton cycles in random graphs
Given an vertex graph whose edges have colored from one of colors
, we define the Hamilton cycle color profile
to be the set of vectors such that there
exists a Hamilton cycle that is the concatenation of paths
, where contains edges. We study
when the edges are randomly colored. We discuss the profile close to the
threshold for the existence of a Hamilton cycle and the threshold for when
.Comment: minor changes reflecting comments from an anonymous refere
An updated survey on rainbow connections of graphs - a dynamic survey
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 introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow -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
Results on monochromatic vertex disconnection of graphs
Let be a vertex-colored graph. A vertex cut of is called a monochromatic vertex cut if the vertices of are colored with the same color. A graph is monochromatically vertex-disconnected if any two nonadjacent vertices of have a monochromatic vertex cut separating them. The monochromatic vertex disconnection number of , denoted by , is the maximum number of colors that are used to make monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: , connectivity and block decomposition. We determine the value of for some well-known graphs, and then characterize when and all blocks of are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph with for any . Finally, we study the ErdΕs-Gallai-type results for , and completely solve them