Turán number and decomposition number of intersecting odd cycles

Given a graph $H$, the Turán function $\mathrm{ex}(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erd?s et al. (1995) determined the Turán function $\mathrm{ex}(n,H_{s,0})$ and the corresponding extremal graphs. Recently, Hou et al. (2016) determined $\mathrm{ex}(n,H_{0,t})$ and the extremal graphs, where the $t$ cycles have the same odd length $q$ with $q?5$. In this paper, we further determine $\mathrm{ex}(n,H_{s,t})$ and the extremal graphs, where $s?0$ and $t?1$. Let $?(n,H)$ be the smallest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E\left(G\right)$ can be partitioned into at most $?(n,H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $?(n,H)=\mathrm{ex}(n,H)$ for $\chi \left(H\right)?3$ and all sufficiently large $n$. Liu and Sousa (2015) verified the conjecture for $H_{s,0}$. In this paper, we further verify Pikhurko and Sousa’s conjecture for $H_{s,t}$ with $s?0$ and $t?1$.     

Sum choice number of generalized θ-graphs

Let $G=\left(VE\right)$ be a simple graph and for every vertex $v\in V$ let $L\left(v\right)$ be a set (list) of available colors. $G$ is called $L$-colorable if there is a proper coloring $\phi$ of the vertices with $\phi \left(v\right)\in L\left(v\right)$ for all $v\in V$. A function $f:V\to \mathbb{N}$ is called a choice function of $G$ and $G$ is said to be $f$-list colorable if $G$ is $L$-colorable for every list assignment $L$ choice function is defined by $size\left(f\right)={\sum }_{v\in V}f\left(v\right)$ and the sum choice number ${\chi }_{sc}\left(G\right)$ denotes the minimum size of a choice function of $G$.Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then.For $r\ge 3$ a generalized ${\theta }_{k_1{k}_2\dots k_r}$-graph is a simple graph consisting of two vertices $v_1$ and $v_2$ connected by $r$ internally vertex disjoint paths of lengths $k_1,k_2,\dots ,k_r$ $(k_1\le k_2\le ?\le k_r)$.In 2014, Carraher et al. determined the sum-paintability of all generalized $\theta$-graphs which is an online-version of the sum choice number and consequently an upper bound for it.In this paper we obtain sharp upper bounds for the sum choice number of all generalized $\theta$-graphs with $k_1\ge 2$ and characterize all generalized $\theta$-graphs $G$ which attain the trivial upper bound $\left|V\left(G\right)\right|+\left|E\left(G\right)\right|$.     

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

On the number of conjugacy classes of π-elements in finite groups

Let G be a finite group and π be a set of primes. Put ${d_{\pi}(G) = k_{\pi}(G)/|G|_{\pi}}$ , where ${k_{\pi}(G)}$ is the number of conjugacy classes of π-elements in G and |G| _π is the π-part of the order of G. In this paper we initiate the study of this invariant by showing that if ${d_{\pi}(G) > 5/8}$ then G possesses an abelian Hall π-subgroup, all Hall π-subgroups of G are conjugate, and every π-subgroup of G lies in some Hall π-subgroup of G. Furthermore, we have ${d_{\pi}(G) = 1}$ or ${d_{\pi}(G) = 2/3}$ . This extends and generalizes a result of W. H. Gustafson.     

Is breaking of ensemble equivalence monotone in the number of constraints?

Breaking of ensemble equivalence between the microcanonical ensemble and the canonical ensemble may occur for random graphs whose size tends to infinity, and is signaled by a non-zero specific relative entropy between the two ensembles. In Garlaschelli et al. (2017) and Garlaschelli et al. (0000) it was shown that breaking occurs when the constraint is put on the degree sequence (configuration model). It is not known what is the effect on the relative entropy when the number of constraints is reduced, i.e., when only part of the nodes are constrained in their degree (and the remaining nodes are left unconstrained). Intuitively, the relative entropy is expected to decrease. However, this is not a trivial issue because when constraints are removed both the microcanonical ensemble and the canonical ensemble change. In this paper a formula for the relative entropy valid for generic discrete random structures, recently formulated by Squartini and Garlaschelli, is used to prove that the relative entropy is monotone in the number of constraints when the constraint is on the degrees of the nodes. It is further shown that the expression for the relative entropy corresponds, in the dense regime, to the degrees in the microcanonical ensemble being asymptotically multivariate Dirac and in the canonical ensemble being asymptotically Gaussian.     

On the number of limit cycles of polynomial Liénard systems

Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Liénard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.     

On the measure of approximation of the number π by algebraic numbers

By means of the method of the Laurent interpolation determinant, it is proved that, if ζ is an algebraic number, the real numbersd andL satisfy the inequalitiesd≥degζ,L≥L(ζ), andL≥3, and the numberd is sufficiently large, then the inequality

The number of eigenvalues of the two-particle discrete Schrödinger operator

We consider the two-particle discrete Schrödinger operator associated with the Hamiltonian of a system of two particles (fermions) interacting only at the nearest neighbor sites. We find the number and the location of the eigenvalues of this operator depending on the particle interaction energy, the system quasimomentum, and the dimension of the lattice ? ^ν , ν ≥ 1.     

On the measurable chromatic number of a space of dimension n ≤ 24

This paper is devoted to the classical problem of finding the measurable chromatic number of n-dimensional Euclidean space, i.e., the value χ _m (? ⁿ ) equal to the least possible number of Lebesgue measurable sets that do not contain pairs of points at a distance of 1 and cover the whole space. Assuming that a certain hypothesis is true, we significantly improve the lower bounds for χ _m (? ⁿ ).     

Average number of iterations of some polynomial interior-point ——Algorithms for linear programming

We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the average number of iterations of these algorithms, coupled with a finite termination technique, is bounded above byO(n ^1.5). The random LP problem is Todd’s probabilistic model with the standard Gauss distribution.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.     

A number of stable matchings in models of the Gale–Shapley type

We give some upper bounds on the maximum number of stable matchings in the Gale–Shapley marriage model with n$n$ men and n$n$ women. We also characterize, with the use of some graph-theoretical notions, the exact number of such matchings, assuming that the preferences of men and women are given.     

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let