On the number of k-dominating independent sets

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot\sqrt[2k]{2}^n$ and $c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c'\cdot1.246^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.Comment: 13 page

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

A result on polynomials derived via graph theory

We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have the same number of roots

Random runners are very lonely

Suppose that $k$ runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least $1/k$ from all the other runners. We prove that, with probability tending to one, a much stronger statement holds for random sets in which the bound $1/k$ is replaced by \thinspace $1/2-\varepsilon$. The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs

Trivial points on towers of curves

We define and study trivial points on towers of curves over number fields, and we show their finiteness in some cases. We relate these to the unboundeness of the gonality of the curves, which we show under some hypothesis. The problem is related to recent results of Cadoret and Tamagawa, and Ellenberg, Hall and Kowalski.Comment: 16 page