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Andrii Arman

Publications and source records attributed to Andrii Arman.

25 records · Page 2Linked to original sources
arXiv2017

Properties of the Fibonacci-sum graph

For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian paths in each $G_n$ have been classified. In this paper, it is shown that each $G_n$ has at most one non-trivial automorphism, which is given explicitly. Other properties of $G_n$ are also found, including the degree sequence, the treewidth, the nature of the bipartition, and that $G_n$ is outerplanar.

math.CO↗
arXiv2017

Equally spaced collinear points in Euclidean Ramsey theory

It is proved that for $k\geq 4$, if the points of $k$-dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two consecutive points. This result is new for $4\leq k\leq 10$.

math.CO↗
arXiv2017

A result in asymmetric Euclidean Ramsey theory

It is proved that if the points of the three-dimensional Euclidean space are coloured in red and blue, then there exist either two red points unit distance apart, or six collinear blue points with distance one between any two consecutive points.

math.CO↗
arXiv2017

The maximum number of cycles in a graph with fixed number of edges

The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant $c$ such that any graph with $m$ edges has at most $(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.

math.CO↗
arXiv2016

Note on Ramsey theorem for posets with linear extensions

In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao.

math.CO↗
arXiv2016

An upper bound for the size of a $k$-uniform intersecting family with covering number $k$

Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. Erdős and Lovász proved that $ \lfloor k! (e-1) \rfloor \leq r(k) \leq k^k.$ Frankl, Ota, and Tokushige improved the lower bound to $r(k) \geq \left( k/2 \right)^{k-1}$, and Tuza improved the upper bound to $r(k) \leq (1-e^{-1}+o(1))k^k$. We establish that $ r(k) \leq (1 + o(1)) k^{k-1}$.

math.CO↗