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Journals and Conferences
We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n n . Bollobás and Riordan  have… (More)
In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice Ln. There the authors gave the following bounds for the asymptotics of f(n), the number of forests of Ln, and α(n), the number of acyclic orientations of Ln: 3.209912 ≤ limn→∞ f(n)1/n2 ≤ 3.84161 and 22/7 ≤ limn→∞ α(n)1/n2 ≤ 3.70925. In this paper we improve… (More)
We define the decision problem data arrangement, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that data arrangement is strongly NP-complete for any fixed d ≥ 2 and explain the… (More)
We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.
The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.
The U -polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them… (More)
We consider the problem of minimizing the diameter of an orientation of a planar graph. A result of Chvátal and Thomassen shows that for general graphs, it is NPcomplete to decide whether a graph can be oriented so that its diameter is at most two. In contrast to this, for each constant l, we describe an algorithm that decides if a planar graph G has an… (More)
We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.
We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k, the characteristic polynomial of any loopless, GF (q)representable matroid with tree-width k has no real zero greater than qk−1.