Small graph classes and bounded expansion

TL;DR: This chapter discusses long paths and Hamiltonicity in Random Graphs, random graphs from Restricted Classes, and lessons on Random Geometric Graphs from Minor-closed Class.

TL;DR: It is found that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and the 2-core which are new even for the uniform (unweighted) case.

TL;DR: The Erdi¾?s-Posa theorem 1965 states that in each graph G which contains at most k disjoint cycles, there is a 'blocking' set B of at most fk vertices such that the graph G - B is acyclic.

TL;DR: In this paper, the authors introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes, and prove the existence of several partition results such as low treewidth and low tree-depth colorings.

TL;DR: In this article, it was shown that for a fixed p, computing the distances between two vertices up to distance p can be computed in constant time per query after a linear time preprocessing.

TL;DR: In this chapter, the main motivation of the authors' work is outlined and it is related to other research.

TL;DR: On the power of linear dependencies, the authors presented a structural approach to the set-sum problem with the Lovasz Local Lemma and the Szemeredi Regularity Lemma.