Rado graph

The Rado graph, as numbered by Ackermann (1937) and Rado (1964).

In the mathematical field of graph theory, the Rado graph, also known as the random graph or the Erdős–Rényi graph, is the unique (up to isomorphism) countably infinite graph R such that for every finite graph G and every vertex v of G, every embedding of G − v as an induced subgraph of R can be extended to an embedding of G into R. As a consequence of this property, the Rado graph contains all finite and countably infinite graphs as induced subgraphs. It is named after Richard Rado, Paul Erdős, and Alfréd Rényi, who all studied it in the early 1960s, but it appears even earlier in the work of Wilhelm Ackermann (1937).

The Rado graph can be constructed by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or by connecting two prime numbers that are congruent to 1 mod 4 when one is a quadratic residue modulo the other. It can also be constructed, with high probability, by a random process on a countably infinite number of vertices in which each pair of vertices has independent probability 1/2 of being connected by an edge.

The first-order logic sentences that are true of the Rado graph are also true of almost all random finite graphs, and the sentences that are false for the Rado graph are also false for almost all finite graphs. In model theory, the Rado graph forms an example of a saturated model of an ω-categorical and complete theory.

History

The Rado graph was first constructed by Ackermann (1937) in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) Erdős & Rényi (1963) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. Richard Rado (1964) rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers essentially equivalent to one of Ackermann's constructions.

Construction via binary numbers

Ackermann (1937) and Rado (1964) constructed the Rado graph using the BIT predicate as follows. They identified the vertices of the graph with the natural numbers 0, 1, 2, ... An edge connects vertices x and y in the graph (where x < y) whenever the xth bit of the binary representation of y is nonzero. Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, because the numbers whose 0th bit is nonzero are exactly the odd numbers. The neighbors of vertex 1 consist of vertex 0 (previously identified because 1 is odd) and all vertices with numbers congruent to 2 or 3 modulo 4 (because those are exactly the numbers with a nonzero bit at index 1), every one of which is greater than 1.

Properties

Extension

The Rado graph satisfies the following extension property: for any finite disjoint sets of vertices U and V, there exists a vertex x connected to everything in U, and to nothing in V. For instance, let

x=2^{1+\max(U\cup V)} + \sum_{u\in U} 2^u.

Then the nonzero bits in the binary representation of x cause it to be adjacent to everything in U. However, x has no nonzero bits in its binary representation corresponding to vertices in V, and x is so large that the xth bit of every element of V is zero. Thus, x is not adjacent to any vertex in V.

Induced subgraphs

This idea of finding vertices adjacent to everything in one subset and nonadjacent to everything in a second subset can be used to build up isomorphic copies of any finite or countably infinite graph G, one vertex at a time. For, let G_i denote the subgraph of G induced by its first i vertices, and suppose that G_i has already been identified as an induced subgraph of a subset S of the vertices of the Rado graph. Let v be the next vertex of G, let U be the neighbors of v in G_i, and let V be the non-neighbors of v in G_i. If x is a vertex of the Rado graph that is adjacent to every vertex in U and nonadjacent to every vertex in V, then S ∪ {x} induces a subgraph isomorphic to G_i + 1.

By induction, starting from the 0-vertex subgraph, every finite or countably infinite graph is an induced subgraph of the Rado graph.

Uniqueness

The Rado graph is, up to graph isomorphism, the only countable graph with the extension property. For, let G and H be two graphs with the extension property, let G_i and H_i be isomorphic induced subgraphs of G and H respectively, and let g_i and h_i be the first vertices in an enumeration of the vertices of G and H respectively that do not belong to G_i and H_i. Then, by applying the extension property twice, one can find isomorphic induced subgraphs G_i + 1 and H_i + 1 that include g_i and h_i together with all the vertices of the previous subgraphs. By repeating this process, one may build up a sequence of isomorphisms between induced subgraphs that eventually includes every vertex in G and H. Thus, by the back-and-forth method, G and H must be isomorphic.^[1]

By applying the same construction to two isomorphic finite subgraphs of the Rado graph, it can be shown that the Rado graph is ultrahomogeneous: any isomorphism between any two induced finite subgraphs of the Rado graph extends to an automorphism of the entire Rado graph.^[1] In particular, there is an automorphism taking any ordered pair of adjacent vertices to any other such ordered pair, so the Rado graph is a symmetric graph.

Robustness against finite changes

If a graph G is formed from the Rado graph by deleting any finite number of edges or vertices, or adding a finite number of edges, the change does not affect the extension property of the graph: for any pair of sets U and V it is still possible to find a vertex in the modified graph that is adjacent to everything in U and nonadjacent to everything in V, by adding the modified parts of G to V and applying the extension property in the unmodified Rado graph. Therefore, any finite modification of this type results in a graph that is isomorphic to the Rado graph.

Partition

For any partition of the vertices of the Rado graph into two sets A and B, or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. Cameron (2001) gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets U_i and V_i that cannot be extended within each subgraph. But then, the union of the sets U_i and the union of the sets V_i would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the complete graph, and the empty graph.^[2] Bonato, Cameron & Delić (2000) and Diestel et al. (2007) investigate infinite directed graphs with the same partition property; all are formed by choosing orientations for the edges of the complete graph or the Rado graph.

A related result concerns edge partitions instead of vertex partitions: for every partition of the edges of the Rado graph into finitely many sets, there is a subgraph isomorphic to the whole Rado graph that uses at most two of the colors. However, there may not necessarily exist an isomorphic subgraph that uses only one color of edges.^[3]

Alternative constructions

One may form an infinite graph in the Erdős–Rényi model by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph has the extension property, and is therefore isomorphic to the Rado graph. This construction also works if any fixed probability p not equal to 0 or 1 is used in place of 1/2. This result, shown by Paul Erdős and Alfréd Rényi in 1963,^[4] justifies the definite article in the common alternative name “the random graph” for the Rado graph: for finite graphs, repeatedly drawing a graph from the Erdős–Rényi model will often lead to different graphs, but for countably infinite graphs the model almost always produces the same graph. Since one obtains the same random process by inverting all choices, the Rado graph is self-complementary.

In one of Ackermann's original 1937 constructions, the vertices of the Rado graph are indexed by the hereditarily finite sets, and there is an edge between two vertices exactly when one of the corresponding finite sets is a member of the other.

As Cameron (2001) describes, the Rado graph may also be formed by a construction resembling that for Paley graphs. Take as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connect two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other (by quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph). Then, for any sets U and V, by the Chinese remainder theorem, the numbers that are quadratic resides modulo every prime in U and nonresidues modulo every prime in V form a periodic sequence, so by Dirichlet's theorem on primes in arithmetic progressions this number-theoretic graph has the extension property.

Model theory and 0-1 laws

Fagin (1976) used the Rado graph to prove a zero–one law for first-order statements in the logic of graphs. The first-order language of graphs is the collection of well-formed sentences in mathematical logic formed from variables representing the vertices of graphs, universal and existential quantifiers, logical connectives, and predicates for equality and adjacency of vertices. For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence

\forall u:\exists v: u\sim v

where the $\sim$ symbol indicates the adjacency relation between two vertices. This sentence $S$ is true for some graphs, and false for others; a graph $G$ is said to model $S$ , written $G\models S$ , if $S$ is true of the vertices and adjacency relation of $G$ .

The extension property of the Rado graph may be expressed by a collection of first-order sentences $E_{i,j}$ , stating that for every choice of $i$ vertices in a set $A$ and $j$ vertices in a set $B$ , all distinct, there exists a vertex adjacent to everything in $A$ and nonadjacent to everything in $B$ . For instance, $E_{1,1}$ can be written as

\forall a:\forall b:a\ne b\rightarrow\exists c:c\ne a\wedge c\ne b\wedge c\sim a\wedge\lnot(c\sim b).

Gaifman (1964) proved that these sentences, together with additional sentences stating that the adjacency relation is symmetric and antireflexive, are the axioms of a complete theory: for each first-order sentence $S$ , exactly one of $S$ and its negation can be proven from these axioms.^[5] Because the Rado graph models the extension axioms, it models all sentences in this theory, and fact that the Rado graph is the unique countable graph with the extension property implies that it is also the unique countable model for this theory. This uniqueness property of the Rado graph can be expressed in model-theoretic terminology by saying that the theory is ω-categorical. Because this theory is categorical and has no finite models, it follows from the Łoś–Vaught test that it is a complete theory.^[6]

As Fagin proved using the compactness theorem, the sentences of this theory (that is, the sentences provable from these axioms and modeled by the Rado graph) are exactly the sentences whose probability of being modeled by a random n-vertex graph (chosen uniformly at random among all graphs on n labeled vertices) approaches one in the limit as n approaches infinity. It follows that every first-order sentence is either almost always true or almost always false for random graphs, and these two possibilities can be distinguished by determining whether the Rado graph models the sentence.^[7] Based on this equivalence, the theory of sentences modeled by the Rado graph has been called "the theory of the random graph" or "the almost sure theory of graphs". However, the computational complexity of determining whether the Rado graph models a given sentence is high: the problem is PSPACE-complete.^[8]

From the model theoretic point of view, the Rado graph is an example of a saturated model.^[9] In this context, a type is a set of variables together with a collection of constraints on the values of some or all of the predicates determined by those variables; a complete type is a type that constrains all of the predicates determined by its variables. A saturated model is a model that realizes all of the types that have a number of variables at most equal to the cardinality of the model. In the theory of graphs, the variables represent vertices and the predicates are the adjacencies between vertices, so a complete type specifies whether an edge is present or absent between every pair of vertices represented by the given variables; that is, it determines the induced subgraph of these vertices. The Rado graph has induced subgraphs of all finite or countably infinite types, so it is saturated.

Related concepts

Although the Rado graph is universal for induced subgraphs, it is not universal for isometric embeddings of graphs, where an isometric embedding is a graph isomorphism which preserves distance. The Rado graph has diameter two, and so any graph with larger diameter does not embed isometrically into it. Moss (1989, 1991) has investigated universal graphs for isometric embedding; he finds a family of universal graphs, one for each possible finite graph diameter. The graph in his family with diameter two is the Rado graph.

The Henson graphs are countable graphs parameterized by a number $i$ , that do not contain an $i$ -vertex clique, and are universal for $i$ -clique-free graphs. They can be constructed as induced subgraphs of the Rado graph.^[10] The Rado graph, the Henson graphs, the disjoint unions of isomorphic cliques, and the complements of disjoint unions of cliques are the only possible countably infinite homogeneous graphs.^[11]

The universality property of the Rado graph can be extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual edge coloring requirement that each color class form a matching. For any finite or countably infinite number of colors χ, there exists a unique countably-infinite χ-edge-colored graph G_χ such that every partial isomorphism of a χ-edge-colored finite graph can be extended to a full isomorphism. With this notation, the Rado graph is just G₁. Truss (1985) investigates the automorphism groups of this more general family of graphs.

Shelah (1984, 1990) investigates universal graphs with uncountably many vertices.

Notes

1 2 Cameron (2001).
↑ Cameron (1990); Diestel et al. (2007).
↑ Pouzet & Sauer (1996).
↑ Erdős & Rényi (1963). Erdős and Rényi use the extension property of graphs formed in this way to show that they have many automorphisms, but do not observe the other properties implied by the extension property. They also observe that the extension property continues to hold for certain sequences of random choices in which different edges have different probabilities of being included.
↑ Gaifman (1964); Marker (2002), Theorem 2.4.2, p. 50.
↑ Marker (2002), Theorem 2.2.6, p. 42.
↑ Fagin (1976); Marker (2002), Theorem 2.4.4, pp. 51–52.
↑ Grandjean (1983).
↑ Lascar (2002).
↑ Henson (1971).
↑ Lachlan & Woodrow (1980).

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