Related papers: The maximum number of minimal codewords in long co…

As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an…

There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let $G$ be a graph with minimum degree at least $k+1$. We prove that if $G$ is…

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$.…

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover ($\operatorname{vc}$) and modular width ($\operatorname{mw}$). We prove that for…

Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an $n$-vertex $r$-graph with density $p \in (0,1)$. The expected number of Hamiltonian cycles in the random $r$-graph model $G_r(n,p)$ is $E(n,p)=p^n(n-1)!/2$ and in the…

Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory…

In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…

In 1980, Gupta, Kahn and Robertson proved that every graph $G$ with minimum degree at least $k\geq 2$ contains a cycle $C$ containing at least $k+1$ vertices each having at least $k$ neighbors in $C$ (so $C$ has at least…

The problem of finding the longest simple cycle in a directed graph is NP-hard, with critical applications in computational biology, scheduling, and network analysis. Existing approaches include exact algorithms with exponential runtimes,…

Recently it was shown (by the author) that every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$ (arXiv:1107.2201v1). In this paper we present the exact analog of this result for…

We show that $3$-graphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing…

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden.…

We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.

A code is called a $q$-query locally decodable code (LDC) if there is a randomized decoding algorithm that, given an index $i$ and a received word $w$ close to an encoding of a message $x$, outputs $x_i$ by querying only at most $q$…

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…

We show that any $2-$factor of a cubic graph can be extended to a maximum $3-$edge-colorable subgraph. We also show that the sum of sizes of maximum $2-$ and $3-$edge-colorable subgraphs of a cubic graph is at least twice of its number of…

We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to…

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $\delta(G)\ge C$ contains a cycle of length $\ell\ge 4$ with $\Omega(\ell/\log^{C}\ell)$ chords for some…

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…