Related papers: A graph-theoretic proof of Cobham's Dichotomy for …

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible…

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*-algebra, there is a corresponding *-automorphism \alpha of another unital C*-algebra such that the two sequences P, P^2,P^3,... and \alpha,…

We consider the problem of detecting a cycle in a directed graph that grows by arc insertions, and the related problems of maintaining a topological order and the strong components of such a graph. For these problems, we give two…

For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a…

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour [JCTB '86] or by a classic result of Chekuri and…

Given a graph $G$ and a non-negative integer $d$ let $\alpha_d(G)$ be the order of a largest induced $d$-degenerate subgraph of $G$. We prove that for any pair of non-negative integers $k>d$, if $G$ is a $k$-degenerate graph, then…

Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple…

In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general…

We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high…

We introduce a new model of correlated randomly growing graphs and study the fundamental questions of detecting correlation and estimating aspects of the correlated structure. The model is simple and starts with any model of randomly…

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and…

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and…

In a recent work, Allen, B\"{o}ttcher, H\`{a}n, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Koml\'{o}s, S\'{a}rk\"{o}zy, and…

We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also…

A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency…

An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes the…

We consider the problem of finding a cycle in a sparse directed graph $G$ that is promised to be far from acyclic, meaning that the smallest feedback arc set in $G$ is large. We prove an information-theoretic lower bound, showing that for…

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of…

We prove the existence of the arithmetic degree for dominant rational self-maps at any point whose orbit is generic. As a corollary, we prove the same existence for \'etale morphisms on quasi-projective varieties and any points on it. We…

Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are $(0,1)$-vectors. We then utilize…