Pair Of Integers Research Articles

Packing cycles in undirected group-labelled graphs

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs (G,γ) where γ assigns to each edge of an undirected graph G an element of an abelian group Γ. As a consequence, we prove that Γ-nonzero cycles (cycles whose edge labels sum to a non-identity element of Γ) satisfy the half-integral Erdős-Pósa property, and we also recover a result of Wollan that if Γ has no element of order two, then Γ-nonzero cycles satisfy the Erdős-Pósa property. As another application, we prove that if m is an odd prime power, then cycles of length ℓmodm satisfy the Erdős-Pósa property for all integers ℓ. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs (ℓ,m).

A NOTE ON REGULAR SETS IN CAYLEY GRAPHS

AbstractA subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$ -regular if R induces a $\kappa $ -regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$ -regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$ -regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$ , $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $ . It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$ -regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$ -regular set of G if and only if it is a $(0,1)$ -regular set of G.

On the degrees of representations of groups not divisible by 2k

Let [Formula: see text] be a finite group. For a given pair of non-negative integers [Formula: see text] and [Formula: see text], we aim to give an explicit formula to count the number of elements of [Formula: see text], the set of all irreducible representations of [Formula: see text] whose degree is not divisible by [Formula: see text]. In this paper, we give formulas for [Formula: see text], when [Formula: see text] is symmetric group, generalized symmetric group, and alternating group. Further, we also have a complete characterization of the elements in [Formula: see text], [Formula: see text], and the self-conjugate partitions of [Formula: see text] and [Formula: see text] with degree not divisible by [Formula: see text].

Some exact results of the generalized Turán numbers for paths

For graphs H and F with chromatic number χ(F)=k, we call H strictly F-Turán-good (or (H,F) strictly Turán-good) if the Turán graph Tk−1(n) is the unique F-free graph on n vertices containing the largest number of copies of H when n is large enough. Let F be a graph with chromatic number χ(F)≥3 and a color-critical edge and let Pℓ be a path with ℓ vertices. Gerbner and Palmer (2020) showed that (P3,F) is strictly Turán good if χ(F)≥4 and they conjectured that (a) this result is true when χ(F)=3, and, moreover, (b) (Pℓ,Kk) is Turán-good for every pair of integers ℓ and k. In the present paper, we show that (H,F) is strictly Turán-good when H is a bipartite graph with matching number ν(H)=⌊|V(H)|2⌋ and χ(F)=3, as a corollary, this result confirms the conjecture (a); we also prove that (Pℓ,F) is strictly Turán-good for 2≤ℓ≤6 and χ(F)≥4, this also confirms the conjecture (b) for 2≤ℓ≤6 and k≥4.

Connected graphs of fixed order and size with maximal Aα-index: The one-dominating-vertex case

For any real number α∈[0,1], by the Aα-matrix of a graph G we mean the matrix Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. The largest eigenvalue of Aα(G) is called the Aα-index of G. Chang and Tam (2011) have proved that for every pair of integers n,k with −1≤k≤n−3, Hn,k, the graph obtained from the star K1,n−1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the Q-index (i.e., the signless Laplacian spectral radius or, equivalently, the A12-index) over all connected graphs with n vertices and n+k edges. In this paper it is proved that for every pair of integers n,k with −1≤k≤n−3, when 12<α<1 or α=12 and k≠2, the graph Hn,k is the unique connected graph that maximizes the Aα-index over all connected graphs with n vertices and n+k edges. This work extends (and also provides an alternative proof for) the above-mentioned result of Chang and Tam. A complete overview of the history of the maximal index problems is also given.

Coloured $$\mathfrak {sl}_r$$ invariants of torus knots and characters of $${\mathcal {W}}_r$$ algebras

Let $$p<p'$$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case of $$\mathfrak {sl}_2$$ , we give a formula for the $$\mathfrak {sl}_r$$ Jones invariants of torus knots $$T (p,p')$$ coloured with the finite-dimensional irreducible representations $$L_r(n\Lambda _1)$$ . When $$r \le p$$ , we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $$L_r(nr\Lambda _1)$$ are essentially the characters of certain minimal model principal $${\mathcal {W}}$$ algebras of type $${{\,\textrm{A}\,}}$$ , namely, $${\mathcal {W}}_r(p,p')$$ , up to some factors independent of p and $$p'$$ but depending on r. In particular, these limits are essentially modular. We expect these limits to be the 0-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for $$p<r$$ .

Knot reversal and rational concordance

AbstractWe give an infinite family of knots that are not rationally concordant to their reverses. More precisely, if denotes the involution of the rational knot concordance group induced by reversal and denotes the subgroup of knots fixed under in , then contains an infinite rank subgroup. As a corollary, we show that there exists a knot such that for every pair of coprime integers and , the ‐cable of is not concordant to the reverse of the ‐cable of .

Modular knots, automorphic forms, and the Rademacher symbols for triangle groups.

É. Ghys proved that the linking numbers of modular knots and the "missing" trefoil in coincide with the values of a highly ubiquitous function called the Rademacher symbol for . In this article, we replace by the triangle group for any coprime pair (p,q) of integers with . We invoke the theory of harmonic Maass forms for to introduce the notion of the Rademacher symbol , and provide several characterizations. Among other things, we generalize Ghys's theorem for modular knots around any "missing" torus knot in and in a lens space.

On a Surface Associated to the Catalan Triangle

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated.

Regular sets in Cayley graphs

In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.

Finding Certain Arithmetic Progressions in 2-Coloured Cyclic Groups

We say a pair of integers (a, b) is findable if the following is true. For any δ > 0 there exists a p0 such that for any prime p ≥ p0 and any red-blue colouring of ℤ/pℤ in which each colour has density at least δ, we can find an arithmetic progression of length a + b inside ℤ/pℤ whose first a elements are red and whose last b elements are blue. Szemeredi’s Theorem on arithmetic progressions implies that (0, k) and (1, k) are find-able for any k. We prove that (2, k) is also findable for any k. However, the same is not true of (3, k). Indeed, we give a construction showing that (3,30000) is not findable. We also show that (14, 14) is not findable.

Hamilton-connected, vertex-pancyclic and bipartite holes

In 2016, McDiarmid and Yolov gave a tight threshold for the existence of Hamilton cycle in graphs with large minimum degree and without large “bipartite hole”(two disjoint sets of vertices with no edge between them) which extends Dirac's classical Theorem. In detail, an (s,t)-bipartite-hole in a graph G consists of two disjoint vertex sets S and T with |S|=s and |T|=t such that E(G[S,T])=∅. Let α˜(G) be the maximum integer such that G contains an (s,t)-bipartite-hole for every pair of nonnegative integers s and t with s+t=r. Motivated by Bondy's metaconjecture, in this paper, we study the existence of vertex-pancyclicity (every vertex is in a cycle of length i for each i∈[3,n] and Hamilton-connectivity(any two vertices can be connected through a Hamilton path). Our central theorem is that for any given μ>0 and sufficiently large n, if G is an n-vertex graph with α˜(G)=μn and δ(G)≥103μn, then G is Hamilton-connected and vertex-pancyclic.

Three numerical approaches to find mutually unbiased bases using Bell inequalities

Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers n,d&amp;#x2265;2 that can be maximally violated in dimension d if and only if n MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the ``four most distant bases'' in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. Lü, B.-G. Englert, {Phys. Rev. A}, { 83} 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.

Gluck twisting roll spun knots

We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an infinite collection of twisted doubles of Gompf's infinite order corks are standard.

Non-Salem Sets in Metric Diophantine Approximation

Abstract A classical result of Kaufman states that, for each $\tau&amp;gt;1$, the set of $\tau $-well approximable numbers $$ \begin{align*} &amp; E(\tau)=\{x \in \mathbb{R}: |xq-r| &amp;lt; |q|^{-\tau} \text{ for infinitely many integer pairs } (q,r)\} \end{align*}$$is a Salem set. A natural question to ask is whether the same is true for the sets of $\tau $-well approximable $n \times d$ matrices when $nd&amp;gt;1$ and $\tau&amp;gt; d/n$. We prove the answer is no by computing the Fourier dimension of these sets. In addition, we show that the set of badly approximable $n \times d$ matrices is not Salem when $nd&amp;gt; 1$. The case of $nd=1$, that is, the badly approximable numbers, remains unresolved.

Split logarithm problem and a candidate for a post-quantum signature scheme

A new form of the hidden discrete logarithm problem, called split logarithm problem, is introduced as primitive of practical post-quantum digital signature schemes, which is characterized in using two non-permutable elements $A$ and $B$ of a finite non-commutative associative algebra, which are used to compute generators $Q=AB$ and $G=BQ$ of two finite cyclic groups of prime order $q$. The public key is calculated as a triple of vectors $(Y,Z,T)$: $Y=Q^x$, $Z=G^w$, and $T=Q^aB^{-1}G^b$, where $x$, $w$, $a$, and $b$ are random integers. Security of the signature scheme is defined by the computational difficulty of finding the pair of integers $(x,w)$, although, using a quantum computer, one can easily find the ratio $x/w\bmod q$.

Disproof of a Conjecture by Woodall on the Choosability of $K_{s,t}$-Minor-Free Graphs

In 2001, in a survey article about list coloring, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable.&#x0D; In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon&gt;0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such that for all integers $s,t $ with $N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\varepsilon)(2s+t)$.

On an Asymptotics of the Number of Representations of a Pair of Integers by Quadratic and Linear Forms with Congruential Condition

On an Asymptotics of the Number of Representations of a Pair of Integers by Quadratic and Linear Forms with Congruential Condition

Generalized signed graphs of large girth and large chromatic number

Assume G is a graph and S is a set of permutations of positive integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e=(u,v) of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,...,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say S is non-trivial if for every positive integer i, there is a permutation π∈S such that π(i)=i, and S is transitive if for every pair (i,j) of positive integers, there is a permutation π∈S such that π(i)=j or π(j)=i. The S-chromatic number of a graph G is the minimum integer k such that any S-labelling (D,σ) of G has a proper k-colouring. This paper constructs, for any non-trivial set S of permutations of integers, for any integers k,g, a graph of girth at least g and S-chromatic number greater than k. We further prove that if S is a non-trivial set of permutations of positive integers, and 2≤k′≤k and g are positive integers, then the following two statements are equivalent: (1) there exists a graph of girth at least g, of chromatic number k′ and S-chromatic number greater than k, (2) for any k′-subset I of [k], there exist a,b∈I and π∈S such that a≠b and either π(a)=b or π(b)=a. As a consequence, there is a bipartite graph of arbitrary large girth and arbitrary large S-chromatic number if and only if S is transitive. In particular, there is a bipartite graph G of arbitrary large girth and arbitrary large group chromatic number.

Families of Laguerre polynomials with alternating group as Galois group

For an arbitrary real number α and a positive integer n, the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined byLn(α)(x)=(−1)n∑j=0n(n+α)(n−1+α)⋯(j+1+α)(n−j)!j!(−x)j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A0∪A∞ of integer pairs (n,α) for which the discriminant of Ln(α)(x) is a nonzero square, where A0 is finite and A∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α≠−1, the only pair (n,α)∈A∞ for which the associated Galois group of Ln(α)(x) is not An is (4,23). In this paper, we verify this conjecture for (α,n)∈A∞ with α∈{−2n,−2n−2,−2n−4}.In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials Ln(α)(x) for n∈N and integers α such that α∈[−2n−4,−2n]. The case α=−2n−1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results.