Related papers: On limits of finite graphs
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…
We prove that for any toric ideal of a graph the degree of any element of Graver basis is bounded above by an exponential function of the maximal degree of a circuit.
In graph theory, knowing the number of complete subgraphs with r vertices that a graph g has, limits the number of its complete subgraphs with s vertices, for s > r. A useful upper bound is provided by the Kruskal-Katona theorem, but this…
If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…
In this short note, we prove the uniqueness of exterior differentiation on locally finite graphs.
We show that the topological space of any infinite graph and its ends is normal. In particular, end spaces themselves are normal.
Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing…
We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.
Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting…
G\'abor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every $\epsilon>0$ there is some finite $k$ such that each graph $G$ in the collection can be broken into connected components of…
We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong…
We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric…
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone…
We prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is $\chi$-bounded.
We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…
A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on…
Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…