Connectivity for bridge-alterable graph classes

A collection of graphs is called bridge-alterable if, for each graph G with a bridge e, G is in the class if and only if G-e is. For example the class of forests is bridge-alterable. For a random forest $F_n$ sampled uniformly from the set of forests on vertex set {1,..,n}, a classical result of Renyi (1959) shows that the probability that $F_n$ is connected is $e^{-1/2 +o(1)}$. Recently Addario-Berry, McDiarmid and Reed (2012) and Kang and Panagiotou (2013) independently proved that, given a bridge-alterable class, for a random graph $R_n$ sampled uniformly from the graphs in the class on {1,..,n}, the probability that $R_n$ is connected is at least $e^{-1/2 +o(1)}$. Here we give a more straightforward proof, and obtain a stronger non-asymptotic form of this result, which compares the probability to that for a random forest. We see that the probability that $R_n$ is connected is at least the minimum over $\frac25 n < t \leq n$ of the probability that $F_t$ is connected.Comment: Amplified the discussion on raising the lower bound 2/5 to 1/

Connectivity for bridge-addable monotone graph classes

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G in A and all subgraphs H of G, H is also in A. We show that for any bridge-addable, monotone class A whose elements have vertex set 1,...,n, the probability that a uniformly random element of A is connected is at least (1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This establishes the special case of a conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotiou (2011).Comment: 11 page

Connectivity for random graphs from a weighted bridge-addable class

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general 'bridge-addable' class of graphs - if a graph is in the class and u and v are vertices in different components then the graph obtained by adding an edge (bridge) between u and v must also be in the class. Various bounds are known concerning the probability of a random graph from such a class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds. For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider 'weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.Comment: 23 page

Random graphs from a weighted minor-closed class

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a more general model. We consider random graphs from a `well-behaved' class of graphs: examples of such classes include all minor-closed classes of graphs with 2-connected excluded minors (such as forests, series-parallel graphs and planar graphs), the class of graphs embeddable on any given surface, and the class of graphs with at most k vertex-disjoint cycles. Also, we give weights to edges and components to specify probabilities, so that our random graphs correspond to the random cluster model, appropriately conditioned. We find that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and we also give results on the 2-core which are new even for the uniform (unweighted) case.Comment: 46 page

Logical limit laws for minor-closed classes of graphs

Let $\mathcal G$ 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 {\em connected} graphs in $\mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $\mathcal G$ 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 length $\approx 5\cdot 10^{-6}$. 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.Comment: minor changes; accepted for publication by JCT

Random graphs from a block-stable class

A class of graphs is called block-stable when a graph is in the class if and only if each of its blocks is. We show that, as for trees, for most $n$-vertex graphs in such a class, each vertex is in at most $(1+o(1)) \log n / \log\log n$ blocks, and each path passes through at most $5 (n \log n)^{1/2}$ blocks. These results extend to `weakly block-stable' classes of graphs

Local Convergence and Stability of Tight Bridge-Addable Graph Classes

A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests. In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class

Random graphs with few disjoint cycles

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B