Béla Bollobás

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The ‘classical’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying(More)
Recently, Barabási and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and(More)
In a previous paper the authors showed that almost all labelled cubic graphs are hamiltonian. In the present paper, this result is used to show that almost all r-regular graphs are hamiltonian for any fixed r ≥ 3, by an analysis of the distribution of 1-factors in random regular graphs. Moreover, almost all such graphs are r-edge-colourable if they have an(More)
We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web.(More)
Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a case-by-case basis. We initiate a more systematic approach. We prove an integrality gap of 2− o(1) for three families of linear relaxations for VERTEX COVER, and our methods seem relevant to other problems as well. ACM Classification:(More)
Recently there has been much interest in studying large-scale real-world networks and attempting to model their properties using random graphs. Although the study of real-world networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ‘smallworld phenomenon’. Since then the main focus of(More)
According to a fundamental result of Erdos and Renyi, the structure of a random graph GM changes suddenly when M n/2: if M = LcnJ and c < 2 then a.e. random graph of order n and since M is such that its 2 largest component has O(log n) vertices, but for c > 2 a.e. GM has a giant component: a component of order (1 ac + o(l))n where a, < 1. The aim of this(More)
Let G1 and GS be graphs with n vertices. If there are edge-disjoint copies of G1 and G, with the same n vertices, then we say there is a packing of G, and Ge . This paper is concerned with establishing conditions on G1 and Ga under which there. is a packing. Our main result (Theorem 1) shows that, with very few exceptions, if G1 and Ga together have at most(More)