相关论文: Order in the chaos with examples from graph theory
Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich…
We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen and extend earlier results on pendant appearances, concerning for example numbers of…
We introduce the notion of recurrence and transience for graphs over non-Archimedean ordered field. To do so we relate these graphs to random walks of directed graphs over the reals. In particular, we give a characterization of the real…
An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed…
We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and a $d$-dimensional random geometric graph $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other…
In the field of complex networks and graph theory, new results are typically tested on graphs generated by a variety of algorithms such as the Erd\H{o}s-R\'{e}nyi model or the Barab\'{a}si-Albert model. Unfortunately, most graph generating…
Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the…
The concept of structural invariance previously introduced by the authors is used to argue that the connection between random matrix theory and quantum systems with a chaotic classical counterpart is in fact largely exact in the…
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all…
In descending generality I survey: five partial orderings of graphs, the induced-subgraph ordering, and examples like perfect, threshold, and mock threshold graphs. The emphasis is on how the induced subgraph ordering differs from other…
Complex systems often exhibit highly structured network topologies that reflect functional constraints. In this work, we investigate how, under varying combinations of system-wide selection rules and special agents, different classes of…
One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the self-embedding monoid, the…
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct…
We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious,…
We study the first order theory of structures over graphs i.e. structures of the form ($\mathcal{G},\tau$) where $\mathcal{G}$ is the set of all (isomorphism types of) finite undirected graphs and $\tau$ some vocabulary. We define the…
End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology. Our main result is that every end-space can topologically be represented by a…
This paper leverages the framework of algorithms-with-predictions to design data structures for two fundamental dynamic graph problems: incremental topological ordering and cycle detection. In these problems, the input is a directed graph…
An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…