Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

Edge-colouring of graphs and hereditary graph properties

summary:Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$, a hereditary graph property ${\mathcal P}$ and $l \ge 1$ we define $\chi '_{{\mathcal P},l}(G)$ to be the minimum number of colours needed to properly colour the edges of $G$, such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in ${\mathcal P}$. We present a necessary and sufficient condition for the existence of $\chi '_{{\mathcal P},l}(G)$. We focus on edge-colourings of graphs with respect to the hereditary properties ${\mathcal O}_k$ and ${\mathcal S}_k$, where ${\mathcal O}_k$ contains all graphs whose components have order at most $k+1$, and ${\mathcal S}_k$ contains all graphs of maximum degree at most $k$. We determine the value of $\chi '_{{\mathcal S}_k,l}(G)$ for any graph $G$, $k \ge 1$, $l \ge 1$, and we present a number of results on $\chi '_{{\mathcal O}_k,l}(G)$

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Thick Forests

We consider classes of graphs, which we call thick graphs, that have their vertices replaced by cliques and their edges replaced by bipartite graphs. In particular, we consider the case of thick forests, which are a subclass of perfect graphs. We show that this class can be recognised in polynomial time, and examine the complexity of counting independent sets and colourings for graphs in the class. We consider some extensions of our results to thick graphs beyond thick forests.Comment: 40 pages, 19 figure