We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension 1 captures precisely the topological cycle space of graphs but works in any dimension.

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π1(G ) with G ⊆ G finite. arXiv:0910.5647v1 [math.CO]

Let S g be the orientable surface of genus g and denote by  g (n, m) the class of all graphs on vertex set [n] = {1, … , n} with m edges embeddable on S g . We prove that the component structure of a graph chosen uniformly at random from  g (n, m) features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős-Rényi random graph G(n, m) chosen uniformly at random from all graphs with vertex set [n] and m edges. It takes place at m = n 2 + O(n 2∕3 ), when the giant component emerges. The second phase transition occurs at m = n + O(n 3∕5 ), when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from G(n, m) and has only been observed for graphs on surfaces.

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G).

Abstract. For integers g, m ≥ 0 and n > 0, let Sg(n, m) denote the graph taken uniformly at random from the set of all graphs on {1, 2, . . . , n} with exactly m = m(n) edges and with genus at most g. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of Sg(n, m), finding that there is often different asymptotic behaviour depending on the ratio m n . In our main results, we show that the probability that Sg(n, m) contains any given non-planar component converges to 0 as n → ∞ for all m(n); the probability that Sg(n, m) contains a copy of any given planar graph converges to 1 as n → ∞ if lim inf m n > 1; the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim inf m n > 1; and the largest face size of Sg(n, m) has a threshold around m n = 1 where it changes from Θ(n) to Θ(ln n) with high probability.