Barak–Erdős graphs and the infinite-bin model

Barak-Erdős graphs and the infinite-bin model

A Barak-Erdős graph is a directed acyclic version of an Erdős-Rényi graph. It is obtained by performing independent bond percolation with parameter p on the complete graph with vertices {1, ..., n}, where the edge between two vertices i < j is directed from i to j. The length of the longest path in this graph grows linearly with the number of vertices, at rate C(p). In this article, we use a co...

The Bin Model

Cops, robbers, and infinite graphs

We study the simplest variant of the cops and robbers game, where one cop tries to catch one robber by moving along the edges of a graph. A well known result connected to this game is that on a finite graph the cop has a winning strategy if and only if the graph is constructible. Chastand et al. proposed the notion of weakly cop-win graphs which they thought could lead to a generalisation of th...

Infinite friendship graphs with infinite parameters

We study infinite graphs in which every set of κ vertices has exactly λ common neighbours. We prove that there exist 2σ such graphs of each infinite order σ if κ is finite and that for κ infinite there are 2λ of them of order λ and none of cardinality greater than λ (assuming the GCH). Further, we show that all a priori admissible chromatic numbers are in fact possible for such graphs.

Language Theory and Infinite Graphs

Automata and language theory study finitely presented mechanisms for generating languages. A language is a family of words. A slight shift in focus is very revealing. Instead of grammars and automata as language generators, one views them as propagators of possibly infinite labelled transition graphs. This is our starting point for pushdown automata. The main goal is to report on a proof of dec...