Mykhaylo Tyomkyn

Walks and Paths in Trees

Recently Csikvari [Combinatorica 30(2) 2010, 125–137] proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend this theorem to all walks. In addition, we give a simpler proof of Csikvaris result and answer one of his questions in the negative. Finally we consider an analogous question for paths rather than walks.

A proof of the rooted tree alternative conjecture

Bonato and Tardif [A. Bonato, C. Tardif, Mutually embeddable graphs and the tree alternative conjecture, J. Combinatorial Theory, Series B 96 (2006), 874-880] conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also make some progress towards the original conjecture for locally finite trees and state some new conjectures.

Universality of Graphs with Few Triangles and Anti-Triangles

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph

Strong Turán stability

{\mathcal G}_{n,1/2}

A Note on the Abelian Sandpile in \pmb{\mathbb{Z}}^{d}

is, in particular, 3-random-like, this can be viewed as a weak version of quasirandomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern. We then show that for larger subgraphs, 3-random-like sequences demonstrate a completely different behaviour. We prove that for every graph

Completing partial packings of bipartite graphs

H

An improved bound for the Manickam-Miklós-Singhi conjecture

on

Distances in Graphs

n\geq R(10,10)

Strong Turán Stability

vertices there exist 3-random-like graphs without an induced copy of

A Turán-Type Problem on Distances in Graphs

H