Small graph classes and bounded expansion

A class of simple undirected graphs is small if it contains at most n!α labelled graphs with n vertices, for some constant α. We prove that for any constants c, ε > 0, the class of graphs with expansion bounded by the function f(r) = c 1/3−ε is small. Also, we show that the class of graphs with expansion bounded by 6 · 3 √ r log(r+e) is not small. We work with simple undirected graphs, without loops or parallel edges. A class of graphs is small if it contains at most n!α different (but not necessarily non-isomorphic) labelled graphs on n vertices, for some constant α. For example, the class of all trees is small, as there are exactly nn−2 < n!e trees on n vertices. Norine, Seymour, Thomas and Wollan [8] showed that all proper minorEmail addresses: [email protected] (Zdeněk Dvořák), [email protected] (Serguei Norine) Supported by the project 1M0021620808 of the Ministry of Education of the Czech Republic. Partially supported by the NSF under Grant DMS-0701033. Preprint submitted to Elsevier June 16, 2009 closed classes of graphs are small, answering the question of Welsh [9]. This question was motivated by the results of McDiarmid, Steger and Welsh [2] regarding random planar graphs. These results in fact hold for any class of graphs that is small and addable. A class G is addable if • G ∈ G if and only if every component of G belongs to G, and • if G1, G2 ∈ G, v1 ∈ V (G1) and v2 ∈ V (G2), then the graph obtained from the disjoint union ofG1 andG2 by adding the edge {v1, v2} belongs to G. Many naturally defined graph classes are addable (for example, minor-closed classes defined by excluding a set of 2-connected minors), and this condition is usually easy to verify. The more substantial assumption thus is that the class is small. Let G be a class that is small and addable, and let N(n) be the number of labelled graphs in G with n vertices. In [2] the following results (among others) were shown: • The limit c = limn→∞(N(n)/n!) exists and is finite. • If K1,k+1 ∈ G, then there exist constants d and n0 such that letting ak = d/(c (k + 2)!), the probability that a random graph in G on n ≥ n0 vertices has fewer than akn vertices of degree k is at most e−akn. Also, a similar result is shown for the number of appearances of arbitrary connected subgraphs. • The probability that a random graph in G on n vertices has an isolated vertex is at least a1/e + o(1) (on the other hand, the probability that 2 such a graph is connected is greater than zero as well). Let us now recall the notion of classes of graphs with bounded expansion, as defined by Nešetřil and Ossona de Mendez [6, 3, 4, 5]. The grad (Greatest Reduced Average Density) with rank r of a graph G is equal to the largest average density of a graph G′ that can be obtained from G by removing some of the vertices (and possibly edges) and then contracting vertex-disjoint subgraphs of radius at most r to single vertices (arising parallel edges are suppressed). The grad with rank r of G is denoted by ∇r(G). In particular, 2∇0(G) is the maximum average degree of a subgraph of G. Given a function f : Z+ → R+, a graph has expansion bounded by f if ∇r(G) ≤ f(r) for every integer r. A class G of graphs has expansion bounded by f if the expansion of every G ∈ G is bounded by f . Finally, we say that a class of graphs G has bounded expansion if there exists a function f such that the expansion of G is bounded by f . The concept of classes of graphs with bounded expansion proves surprisingly powerful. Many classes of graphs have bounded expansion (proper minor-closed classes, classes of graphs with bounded maximum degree, classes of graphs excluding subdivision of a fixed graph, . . . ), and many results for proper minor-closed classes (existence of colorings, small separators, light subgraphs, . . . ) generalize to classes of graphs with bounded expansion (possibly with further natural assumptions). The classes of graphs with bounded expansion are also interesting from the algorithmic point of view, as the proofs of the mentioned results usually give simple and efficient algorithms. Furthermore, fast algorithms and data structures for problems like deciding whether a graph contains a fixed subgraph, or for determining the distance