Rainbow Connection of Random Regular Graphs

An edge colored graph $G$ 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 $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connection of the random $r$-regular graph $G=G(n,r)$ of order $n$, where $r\ge 4$ is a constant. We prove 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

Colorful Hamilton cycles in random graphs

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges. We study $hcp(G_{n,p})$ 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 $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r:m_1+m_2+\cdots+m_r=n\}$.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 $k$-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

Results on monochromatic vertex disconnection of graphs

Let $G$ be a vertex-colored graph. A vertex cut $S$ of $G$ is called a monochromatic vertex cut if the vertices of $S$ are colored with the same color. A graph $G$ is monochromatically vertex-disconnected if any two nonadjacent vertices of $G$ have a monochromatic vertex cut separating them. The monochromatic vertex disconnection number of $G$, denoted by $mvd(G)$, is the maximum number of colors that are used to make $G$ monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: $mvd(G)$, connectivity and block decomposition. We determine the value of $mvd(G)$ for some well-known graphs, and then characterize $G$ when $n-5\leq mvd(G)\leq n$ and all blocks of $G$ are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph $G$ with $mvd(G) = k$ for any $k$. Finally, we study the Erdős-Gallai-type results for $mvd(G)$, and completely solve them