On automorphisms of the countable p.e.c. graph

Graphs and Combinatorics, 1989

Let X be a connected locally finite graph with vertex-transitive automorphism group. If X has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup of AUT(X) and the action of AUT(X) on V(X) is imprimitive if X is not finite. If X has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.

European Journal of Combinatorics, 2010

The paper contains a construction of a universal countable graph, different from the Rado graph, such that for any of its vertices both the neighbourhood and the non-neighbourhood induce subgraphs isomorphic to the whole graph. This solves an open problem proposed by A. Bonato; see Problem 20 in Cameron [5]. We supply a construction of several non-isomorphic graphs with the property, and consider tournaments with an analogous property.

Journal of The London Mathematical Society-second Series, 1971

In [1, 2] R. Rado proved several theorems on universal graphs. It is the purpose of this note to point out that the substance of these theorems, in fact strengthenings of them, can be obtained from general algebraic results of B. Jonsson or from related model theoretic results of M. Morley and R. Vaught. However, as we explain below, it seems to be the case that not all Rado's results can be obtained in this way, a fact which gives a little added interest to the comparison of results.

Mathematica Slovaca

Let XY L,T consist of all countable L-structures M that satisfy the axioms T and in which all homomorphisms of type X (these could be plain homomorphisms, monomorphisms, or isomorphisms) between finite substructures of M are restrictions of an endomorphism of M of type Y (for example, an automorphism or a surjective endomorphism). Lockett and Truss introduced 18 such classes for relational structures. For a given pair L, T however, two or more morphism-extension properties may define the same class of structures. In this paper, we establish all equalities and inequalities between morphism-extension classes of countable graphs.

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain complete or empty induced subgraphs of size at least |V(G)|^c(H), then H can be isomorphically embedded into G ? The positive answer has become known as the Erd\H{o}s-Hajnal conjecture. In Theorem 3.13 of the present paper we settle this conjecture in the affirmative. To do so, we are studying here the fine structure of ultraproducts of finite sets, so our investigations have a model theoretic character.

European Journal of Combinatorics, 2003

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2 ω the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended ("partnership theorem"). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs (where we solve a problem posed in [19]) and to the density problem for countable graphs.

Doklady Mathematics, 2011

Discrete Mathematics, 1997

In this paper we investigate the closure ~-* under substitution-composition of a family of graphs ~,, defined by a set Lr of forbidden configurations. We first prove that ~-* can be defined by a set L~* of forbidden subgraphs. Next, using a counterexample we show that ~* is not necessarily a finite set, even when .~ is finite. We then give a sufficient condition for ~* to be finite and a simple algorithm for enumerating all the graphs of ~.* As application, we obtain new classes of perfect graphs.

In this paper, we deduce some properties of f -sets of connected graphs. Also, we introduce the concept of fixing share of each vertex of a fixing set D to see the participation of each vertex when fixing a connected graph G. We define a parameter, called the fixing percentage, by using the concept of fixing share, which is helpful in determining the measure of the amount of fixing done by the elements of D in G.

Discrete Mathematics, 1999

A corrected proof is given for the existence of a universal countable {Cj, CS,. , Czii ,)-free graph. We also prove that there is a universal countable w-free graph. There is no universal countable H-free graph if H is the disjoint union of 3 or more complete n-cliques for some n 22, plus one vertex, joined to every other point.