Related papers: On unavoidable obstructions in Gaussian walks

Let $n \ge 5$ and $k\ge 4$ be positive integers. We determine the maximum size of digraphs of order n that avoid distinct walks of length k with the same endpoints. We also characterize the extremal digraphs attaining this maximum number…

We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above…

The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (s_i,t_i) contains paths P_1,...,P_k such that P_i connects s_i and t_i for i=1,...,k, and P_i and P_j have neither common vertices nor…

Let $m>1$ be an integer, and let $I(\mathbb{Z}_m)^*$ be the set of all non-zero proper ideals of $\mathbb{Z}_m$. The intersection graph of ideals of $\mathbb{Z}_m$, denoted by $G(\mathbb{Z}_m)$, is a graph with vertices $I(\mathbb{Z}_m)^*$…

We investigate the avoidability of unary patterns of size of four with morphic permutations. More precisely, we show that, for the positive integers $i,j,k$, the sizes of the alphabets over which a pattern $x \pi ^ {i} (x) \pi^{j}(x)…

This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…

We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is…

Let $c_n = c_n(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $\mathbb{Z}^d$. Let $\mu = \lim_n c_n^{1/n}$ denote the connective constant of $\mathbb{Z}^d$. In…

For two strings u, v over some alphabet A, we investigate the problem of embedding u into w as a subsequence under the presence of generalised gap constraints. A generalised gap constraint is a triple (i, j, C_{i, j}), where 1 <= i < j <=…

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the…

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all…

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a…

In 1989, Erd\H{o}s conjectured that for a sufficiently large $n$ it is impossible to place $n$ points in general position in a plane such that for every $1\le i \le n-1$ there is a distance that occurs exactly $i$ times. For small $n$ this…

Extending the methods of Metrebian (2018), we prove that any symmetric punctured interval tiles $\mathbb Z^3$. This solves two questions of Metrebian and completely resolves a question of Gruslys, Leader and Tan. We also pose a question…

In this paper, we mainly concerned about deriving the general formula to count the possible positions of $n$ step random walk in $\mathbb{Z}^d$ with unit length in each step, which we denoted as $|P_n^{d}|$. For our results, we firstly…

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a…

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. Applying probabilistic methods, an upper bound for the…