Let G be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G on n vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface S. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of the same length ≈ 5.39 · 10 −6 . Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on n vertices, even in FO.

Abstract. Two graphs G and H are hypomorphic if there exists a bijection ϕ :

Abstract. The conjecture of Bollobás and Komlós, recently proved by Bött-cher, Schacht, and Taraz [Math. Ann. 343(1), 175-205, 2009], implies that for any γ > 0, every balanced bipartite graph on 2n vertices with bounded degree and sublinear bandwidth appears as a subgraph of any 2n-vertex graph G with minimum degree (1 + γ)n, provided that n is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of ( 1 2 + γ)n when we have the additional structural information of the host graph G being balanced bipartite.This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladký and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for Kr,s-factors, with r and s fixed. Moreover, it implies that the set of Hamilton cycles of G is a generating system for its cycle space.

Two graphs G and H are hypomorphic if there exists a bijection ϕ :Nash-Williams proved that all locally finite connected graphs with a finite number ě 2 of ends are reconstructible, and asked whether locally finite connected graphs with one end or countably many ends are also reconstructible.In this paper we construct non-reconstructible connected graphs of bounded maximum degree with one and countably many ends respectively, answering the two questions of Nash-Williams about the reconstruction of locally finite graphs in the negative.2010 Mathematics Subject Classification. 05C60, 05C63.

Abstract. For a graph X, let f 0 pXq denote its number of vertices, δpXq its minimum degree and Z 1 pX; Z{2q its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z{2-coefficients). Call a graph Hamiltongenerated if and only if the set of all Hamilton circuits is a Z{2-generating system for Z 1 pX; Z{2q. The main purpose of this paper is to prove the following: for every γ ą 0 there exists n 0 P Z such that for every graph X with f 0 pXq ě n 0 vertices, (1) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is odd, then X is Hamilton-generated, (2) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z 1 pX; Z{2q, and the set of all circuits of X having length either f 0 pXq´1 or f 0 pXq generates all of Z 1 pX; Z{2q, (3) if δpXq ě p 1 4`γ qf 0 pXq and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.