We give an overview of recent advances on the graph isomorphism problem. Our main focus will be on Babai's quasi-polynomial time isomorphism test and subsequent developments that led to the design of isomorphism algorithms with a quasi-polynomial parameterized running time of the from $n^{\polylog k}$, where $k$ is a graph parameter such as the maximum degree. A second focus will be the combinatorial Weisfeiler-Leman algorithm.
The Johnson graph J (v, k) has, as vertices, all k-subsets of a v-set V, with two k-subsets adjacent if and only if they share k 1 common elements of V. Subsets of vertices of J (v, k) can be interpreted as the blocks of an incidence structure, or as the codewords of a code, and automorphisms of J (v, k) leaving the subset invariant are then automorphisms of the corresponding incidence structure or code. This approach leads to interesting new designs and codes. For example, numerous actions of the Mathieu sporadic simple groups give rise to examples of Delandtsheer designs (which are both flag-transitive and anti-flag transitive), and codes with large minimum distance (and hence strong error-correcting properties). The paper surveys recent progress, explores links between designs and codes in Johnson graphs which have a high degree of symmetry, and discusses several open questions.
This paper surveys what is currently known about the maximal sub-groups of the finite simple groups. After briefly introducing the groups themselves, if their maximal subgroups are completely determined then we present this classification. For the remaining finite simple groups our current knowledge is only partial: we describe the state of play, as well as giving some results that apply more generally. We also direct the reader towards computational resources for the construction of maximal subgroups. After this, we present three sample applications, selected because they combine group theoretical and combinatorial arguments, and because they use either or both of the detailed classifications and the looser statements that can be made about all maximal subgroups. In particular, we discuss results relating to generation, and the generating graph; results concerning bases; and some applications to computational complexity, in particular to graph colouring and other problems with no known polynomial-time solution.
We will review various ways to decompose the edges of a graph into few simple substructures. We will mainly focus on variants of edge colouring, and discuss specifically the discharging method and re-colouring techniques. All graphs considered here are simple and finite, unless specified otherwise. We assume the reader has some familiarity with elementary notions in graph theory. This chapter is meant as a walkthrough of two techniques that can be useful and that are perhaps not as widely known as they could be. The first one is the discharging method, which makes up for most of this chapter (Sections 2 to 5), and the second one is the use of re-colouring (Section 6).
Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which are similar in some way. One way to do this is to take a sample of several random graphs from the family, to gather information about what is "typical". Hence there is a need for algorithms which can generate graphs uniformly (or approximately uniformly) at random from the given family. Since a large sample may be required, the algorithm should also be computationally efficient. Rigorous analysis of such algorithms is often challenging, involving both combinatorial and probabilistic arguments. We will focus mainly on the set of all simple graphs with a particular degree sequence, and describe several different algorithms for sampling graphs from this family uniformly, or almost uniformly.
