Universal Graphs with Forbidden Subgraphs and Algebraic Closure

Posets of copies of countable ultrahomogeneous tournaments

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TK n ) and ℱ(HK n ),n≦ℵ0, contain a universal element. In fact, for ℱ(TK 4)=ℱ(HK 4) a strongly universal graph is constructed, whereas for 5≦n≦ℵ0 the classes ℱ(TK n ) and ℱ(HK n ) have no universal elements.

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph of R). A brief overview of known universality results for some induced-hereditary subsets of Ic is provided. We then construct a k-degenerate graph which is universal for the induced-hereditary property of finite k-degenerate graphs. In order to attempt the corresponding problem for the property of countable graphs with colouring number at most k + 1, the notion of a property with assignment is introduced and studied. Using this notion, we

On the order of countable graphs

Let Forb(G) denote the class of graphs with countable vertex sets which do not contain G as a subgraph. If G is finite, 2-connected, but not complete, then Forb(G) has no element which contains every other element of Forb(G) as a subgraph, i.e., this class contains no universal graph. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 53–58, 1997

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P . Let P1,P2, . . . ,Pn be graph properties of finite character, a graph G is said to be (uniquely) (P1,P2, . . . ,Pn)partitionable if there is an (exactly one) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. Let us denote by R = P1◦P2◦ · · · ◦Pn the class of all (P1,P2, . . . ,Pn)-partitionable graphs. A property R = P1◦P2◦ · · · ◦Pn, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P1,P2, . . . ,Pn)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property Pi has a weakly universal graph. 242 J. Bucko and P. Mihok

Computation of the degree of rational surface parametrizations