Generation of full cycles by a composition of NLFSRs

Non-linear feedback shift registers (NLFSRs) are a generalization of linear feedback shift registers in which a current state is a non-linear function of the previous state. The interest in NLFSRs is motivated by their ability to generate pseudo-random sequences which are typically hard to break with existing cryptanalytic methods. However, it is still not known how to construct large \(n\) -stage NLFSRs which generate full cycles of \(2^n\) possible states. This paper presents a method for generating full cycles by a composition of NLFSRs. First, we show that an \(n*k\) -stage register with period \(O(2^{2n})\) can be constructed from \(k\) NLFSRs with \(n\) -stages by adding to their feedback functions a logic block of size \(O(nk)\) , for \(k > 1\) . This logic block implements Boolean functions representing pairs of states whose successors have to be exchanged in order to join cycles. Then, we show how to join all cycles into one by using one more logic block of size \(O(nk)\) .     

The extended coset leader weight enumerator of a twisted cubic code

Designs, Codes and Cryptography - The extended coset leader weight enumerator of the generalized Reed–Solomon $$[q+1,q-3,5]_q$$ code is computed. In this computation methods in finite...     

Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

Double coset density in classical algebraic groups

We classify all pairs of reductive maximal connected subgroups of a classical algebraic group that have a dense double coset in . Using this, we show that for an arbitrary pair of reductive subgroups of a reductive group satisfying a certain mild technical condition, there is a dense -double coset in precisely when is a factorization.

     

The confirmation of a conjecture on disjoint cycles in a graph

Let $t$ and $k$ be two integers with $t\ge 5$ and $k\ge 2$. For a graph $G$ and a vertex $x$ of $G$, we use $d_G\left(x\right)$ to denote the degree of $x$ in $G$. Define ${\sigma }_t\left(G\right)$ to be the minimum value of ${\sum }_{x\in X}{d}_G\left(x\right)$, where $X$ is an independent set of $G$ with $\left|X\right|=t$. This paper proves the following conjecture proposed by Gould et al. (2018). If $G$ is a graph of sufficiently large order with ${\sigma }_t\left(G\right)\ge 2kt?t+1$, then $G$ contains $k$ vertex-disjoint cycles.     

Every connected regular graph of even degree is a Schreier coset graph

Using Petersen's theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.     

The heterochromatic cycles in edge-colored graphs

Given a graph and an edge coloring C of G, a heterochromatic cycle of G is a cycle in which any pair of edges have distinct colors. Let d ^c (v), named the color degree of a vertex v, be the maximum number of distinct colored edges incident with v. In this paper, we give several sufficient conditions for the existence of heterochromatic cycles in edge-colored graphs.     

Disjoint long cycles in a graph

We prove that if G is a graph of order at least 2k with k ? 9 and the minimum degree of G is at least k + 1, then G contains two vertex-disjoint cycles of order at least k. Moreover, the condition on the minimum degree is sharp.     

Two vertex-disjoint cycles in a graph

LetG be a graph of ordern 6 with minimum degree at least (n + 1)/2. Then, for any two integerss andt withs 3,t 3 ands + t n, G contains two vertex-disjoint cycles of lengthss andt, respectively, unless thatn, s andt are odd andG is isomorphic toK _{(n–1)/2,(n–1)/2} + K₁. We also show that ifG is a graph of ordern 8 withn even and minimum degree at leastn/2, thenG contains two vertex-disjoint cycles with any given even lengths provided that the sum of the two lengths is at mostn.     

Vertex-disjoint chorded cycles in a graph

In this paper we prove: Let k≥1 be an integer and G be graph with at least 4k vertices and minimum degree at least ⌊7k/2⌋. Then G contains k vertex-disjoint cycles such that each of them has at least two chords in G.     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

The threshold for hamilton cycles in the square of a random graph

We show in G_n.p' that the threshold for δ(G) ? 1 is the threshold for G²–G to be Hamiltonian. © 1994 John Wiley & Sons, Inc.     

The boundary of a smooth set has full Hausdorff dimension

We prove that if the restriction of the Lebesgue measure to a set A⊂[0,1] with 0<|A|<1 is a smooth measure, then the boundary of A must have full Hausdorff dimension.