N. Beluhov

activity

20132026

collaborators

13 papers

math.NT2026

Diamond Determinants and Somos Sequences

Nikolai Beluhov

A Somos sequence of order $n$ is defined by a quadratic recurrence of width $n + 1$ . Some of the remarkable properties of these sequences for small $n$ are tied to certain matrices…

math.CO2025

Minimum-Turn Tours of Even Polyominoes

Nikolai Beluhov

Let $P$ be a connected bounded region in the plane formed out of $2 \times 2$ blocks joined by their sides. Peng and Rascoussier conjectured that all minimum-turn Hamiltonian cycle…

math.CO2024

Double Tiles

Nikolai Beluhov

Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomo…

math.CO2023

Slitherlink Signatures

Nikolai Beluhov

Let $G$ be a planar graph and let $C$ be a cycle in $G$ . Inside of each finite face of $G$ , we write down the number of edges of that face which belong to $C$ . This is the signatur…

math.CO2023

Snake Paths in King and Knight Graphs

Nikolai Beluhov

A snake path in a graph $G$ is a path in $G$ which is also an induced subgraph of $G$ . For all $n$ , we find the greatest length of a snake path in the $n \times n$ king graph and w…

math.CO2021

Induction on Descent in Leaper Graphs

Nikolai Beluhov

We construct an infinite ternary tree $L$ whose root is the knight and whose vertices are all skew free leapers. We define the descent of a skew free leaper to be its "a…

math.CO2021

One Curious Identity Counting Graceful Labelings

Nikolai Beluhov

Let $a$ and $b$ be positive integers with prime factorisations $a = p_{1}^{n} p_{2}^{n}$ and $b = q_{1}^{n} q_{2}^{n}$ . We prove that the number of essentially distinct $α$ -graceful labelings of the…

math.CO2019

The Flexibility and Rigidity of Leaper Frameworks

Nikolai Beluhov

A leaper framework is a bar-and-joint framework whose joints are integer points forming a rectangular grid and whose bars correspond to all moves of a given leaper within that grid…

math.CO2017

A Proof of Willcocks's Conjecture

Nikolai Beluhov

We give a proof of Willcocks's Conjecture, stating that if $p - q$ and $p + q$ are relatively prime, then there exists a Hamiltonian tour of a $(p, q)$ -leaper on a square chessboar…

math.CO2017

The Second Leaper Theorem

Nikolai Beluhov

A $(p, q)$ -leaper is a fairy chess piece that, from a square $a$ , can move to any of the squares $a + (\pm p, \pm q)$ or $a + (\pm q, \pm p)$ . Let $L$ be a $(p, q)$ -leaper with $p…

math.CO2016

Renyi-Ulam Games and Forbidden Substrings

Nikolai Beluhov

The Renyi-Ulam game is played between two players, the Seeker and the Obscurer. The Obscurer thinks of a number between 1 and $n$ . The Seeker wishes to identify that number. On eac…

math.CO2013

On Crosspatch Knight's Tours

Nikolai Beluhov

A knight's tour is often represented as a broken line connecting the centers of successively visited squares. We say that two knight moves form a cross if the midpoints of their re…

math.MG2013

The Mixed Poncelet-Steiner Closure Theorem

Nikolai Beluhov

We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning coni…