The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

/pdf/the-probabilistic-method-5g63k0itke.pdf

The Probabilistic Method

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

/pdf/a-dynamic-survey-of-graph-labeling-198znmwxtq.pdf

A Dynamic Survey of Graph Labeling

Distance-Regular Graphs

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are also presented, as well as various generalisations to random graphs with given degree sequence. A major feature in this area is the small subgraph conditioning method. When applicable, this establishes a relationship between random regular graphs with uniform distribution, and non-uniform models of random regular graphs in which the probability of a graph G is weighted according to the number of subgraphs G has of a certain type. Information can be obtained in this way on the probability of existence of various types of spanning subgraphs, such as Hamilton cycles and decompositions into perfect matchings. Uniformly distributed labelled random regular graphs receive most of the attention, but also included are several non-uniform models which come about in a natural way. Some of these appear as spin-offs from the small subgraph conditioning method, and some arise from algorithms which use simple approaches to generating random regular graphs. A quite separate role played by algorithms is in the derivation of random graph properties by analysing the performance of an appropriate greedy algorithm on a random regular graph. Many open problems and conjectures are given.

Surveys in Combinatorics, 1999: Models of Random Regular Graphs

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

http://emis.matem.unam.mx/journals/EJC/Surveys/ds16.pdf

Dynamic Cage Survey

It is possible to define many variations of packing and covering invariants for graphs which involve paths and cycles. These can be given terminology which is sufficiently intuitive that one can remember the definitions, e.g., arboricity, linear arboricity, point arboricity, point linear arboricity, anarboricity, path number, unpath number, point anarboricity, and cyclicity. Most of these concepts are fundamental but it is not easy to determine the value of these invariants for general graphs. We investigate these concepts and relations among them for specific families of graphs. In particular, we determine them for complete graphs, complete bipartite graphs, and their line graphs.

Covering and packing in graphs. III: Cyclic and acyclic invariants

The linear arboricity of a graph is the minimum number of linear forests into which its lines can be decomposed. We find that the linear arboricity of every 4-regular graph is 3. This result enables us to obtain bounds for the linear arboricity of any graph in terms of its maximum degree.

Covering and packing in graphs IV: Linear arboricity

A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self-complementarity and regularity.

Paley graphs satisfy all first‐order adjacency axioms

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for $R_k(4)$ and $R_k(5)$ for some small $k$, including $415 \le R_3(5)$, $634 \le R_4(4)$, $2721 \le R_4(5)$, $3416 \le R_5(4)$ and $26082 \le R_5(5)$. Most of the new lower bounds are consequences of general constructions.

/pdf/constructive-lower-bounds-on-classical-multicolor-ramsey-3pb3fl6481.pdf

Geoffrey Exoo

Papers

Dynamic Cage Survey

Covering and packing in graphs. III: Cyclic and acyclic invariants

Covering and packing in graphs IV: Linear arboricity

Paley graphs satisfy all first‐order adjacency axioms

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers