Let H be some fixed graph. We call a graph Gvicarious for H if G is maximal H-free and, for every edge e of G, there is an edge f not in G such that G − e + f is also H-free. We demonstrate various properties of vicarious graphs and several examples are given. It is conjectured that a graph of order n which is vicarious for K3 has size at most (1/4 + o(1))().

On page 551 the first author name should be J. A. De Loera (NOT J. A Loera)

The correct citation for this paper is

J. A. De Loera, J. Lee, S. Margulies and S. Onn (2009) Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz. Combinatorics, Probability and Computing, 18 (4) July, 551–582 doi:10.1017/S0963548309009894, Published online by Cambridge University Press 28 April 2009.

The paper presents yet another way to reformulate the Four Colour Conjecture as a statement concerning conditional probabilities of certain events involving planar graphs.

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.