Discrete duality principle in different random graph models

For a given random graph, a connected component that contains a finite fraction of the entire graph's vertices is called giant. The study of these components started with the \ER model, where it has been proven that removing the (unique) giant component from a random graph is essentially equivalent to another random graph in the same model with different known parameters. This is called the discrete duality principle. In this report we aim at presenting this principle in its historical \ER settings, and then to present more recent generalisations made on random graphs with given degree sequences

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

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present