Béla Bollobás

A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs

Let Δ and n be natural numbers such that Δn = 2m is even and Δ ⩽ (2 log n )1/2 - 1. Then as n →, the number of labelled Δ-regular graphs on n vertices is asymptotic to e − λ − λ 2 ( 2 m ) ! m ! 2 m ( Δ ! ) m where λ = (Δ -1)/2. As a consequence of the method we determine the asymptotic distribution of the number of short cycles in graphs with a given degree sequence, and give analogous formulae for hypergraphs.

The chromatic number of random graphs

AbstractFor a fixed probabilityp, 0<p<1, almost every random graphGn,p has chromatic number

Graph‐theoretic parameters concerning domination, independence, and irredundance

\left( {\frac{1}{2} + o(1)} \right)\log (1/(1 - p))\frac{n}{{\log n}}

Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions

,

The isoperimetric number of random regular graphs

Let 0 p G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p , independently of all other edges. In other words the random variables e ij , 1 ≤ i j , defined by are independent r.v.s with P ( e ij = 1) = p , P ( e ij = 0) = 1 − p . Denote by G n the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, G n will be investigated throughout the note . As in (1), denote by K r a complete graph with r points and denote by k r ( H ) the number of K r s in a graph H . A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of k r ( H ) provided H has n points and m edges. In this note we shall look at the random variables the number of K r s in G n , and the maximal size of a clique in G n .

Reliable density estimates for coverage and connectivity in thin strips of finite length

A vertex x in a subset X of vertices of an undericted graph is redundant if its closed neighbourhood is contained in the union of closed neighborhoods of vertices of X – {x}. In the context of a communications network, this means that any vertex that may receive communications from X may also be informed from X – {x}. The irredundance number ir (G) is the minimum cardinality taken over all maximal sets of vertices having no redundancies. The domination number γ(G) is the minimum cardinality taken over all dominating sets of G, and the independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of vertices of G. The paper contians results that relate these parameters. For example, we prove that for any graph G, ir (G) > γ(G)/2 and for any grpah Gwith p vertices and no isolated vertices, i(G) ≤ p-γ(G) + 1 - ⌈(p - γ(G))/γ(G)⌉.

List-colourings of graphs

How small can the diameter be made by adding a matching to an n-cycle? In this paper this question is answered by showing that the graph consisting of an n-cycle and a random matching has diameter about

Random Graphs of Small Order

\log _2 n