Related papers: Around Pelikan's conjecture on very odd sequences
A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and…
We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…
The Kolakoski sequence is the unique infinite sequence with values in $\{1, 2\}$ and first term twems $1, 2, \ldots$ which equals the sequence of run-lengths of itself, we call this $K(1, 2).$ We define $K(m, n)$ similarly for $m+n$ odd. A…
On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial…
Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer larger than $a_m$ that can be written as the sum of two (not necessarily distinct)…
The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such…
An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyarfas made the conjecture that for all t there exists n such that every graph with no K_t subgraph and no odd hole is n-colourable. We…
Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were…
In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$…
In 1999 Lyngs{\o} and Pedersen proposed a conjecture stating that every binary circular word of length $n$ with equal number of zeros and ones has an antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No progress over a…
Twenty years ago Bondy and Vince conjectured that for any nonnegative integer $k$, except finitely many counterexamples, every graph with $k$ vertices of degree less than three contains two cycles whose lengths differ by one or two. The…
A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization…
Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrimsson. In this paper, we consider ascent sequences of…
In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can…
How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$…
Negative orientable sequences, i.e. periodic sequences with elements from a finite alphabet of size at least three in which an n-tuple or the negative of its reverse appears at most once in a period of the sequence, were introduced by…
In 1984, Plesn\'{i}k determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the…
Professor Cadogan at the University of the West Indies identified special starting points that yield long subsequences where the normalization constant, k, is always one. I studied these special sequences and found an implicit mixed integer…
In 1984, Plesn\'{i}k determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the…
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some…