Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

1. (1)
   There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

   $\varphi\,:\, C\,\longrightarrow\, X$

   a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.
    
2. (2)
   The variety X admits an étale covering by an abelian variety.
    
3. (3)
   The dimension dim X ≤ 1.
    

We prove that the following classes are among those that are of the above type.

• All X with a finite fundamental group.
• All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.
• All X such that the canonical line bundle K _X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.
• All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).

     

Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)

1. (1)
   The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.
    
2. (2)
   There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).
    
3. (3)
   There exists a unique limit model of size \(\lambda \).
    

Our proofs use independence calculus and a generalization of averages to this non first-order context.

     

On a problem of P. Hall for Engel words

Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ ^*(G), respectively. The following problems are generally attributed to P. Hall:

1. (I)
   If π is a set of primes and |G : θ ^*(G)| is a finite π-group, is θ(G) also a finite π-group?
    
2. (II)
   If θ(G) is finite and G satisfies maximal condition on its subgroups, is |G : θ ^*(G)| finite?
    
3. (III)
   If the set \({\{\theta(g_1,\ldots,g_n) \;|\; g_1,\ldots,g_n\in G\}}\) is finite, does it follow that θ(G) is finite?
    

We investigate the case in which θ is the n-Engel word e _n  = [x, _n y] for \({n\in\{2,3,4\}}\) .

     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

Relations between the $${\mathcal {}}$$-ultrafilters

Under CH we show the following results:

1. (1)
   There is a discrete ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
2. (2)
   There is a \(\sigma \)-compact ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
3. (3)
   There is a \({\mathcal {J}}_{\omega ^{3}}\)-ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    

     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Non-singular Solutions of Normalized Ricci Flow on Noncompact Manifolds of Finite Volume

The main result of this paper shows that if g(t) is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold M of finite volume, then the Euler characteristic number χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points \(\{p_{k,l}\}_{l=1}^{N}\) and domains \(\{U_{k,l}\}_{l=1}^{N}\) with p _k,l∈U _k,l satisfying the following:

1. (i)
   \(\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty\) as k→∞, for any fixed l₁≠l₂;
    
2. (ii)
   for each l, (U _k,l,g(t _k ),p _k,l) converges in the \(C_{\mathrm{loc}}^{\infty}\) sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;
    
3. (iii)
   \(\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0\) as k→∞.
    

     

Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in $$L^\infty $$

Consider the supremal functional

$$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$

(1)

applied to \(W^{1,\infty }\) maps \(u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N\), \(N\ge 1\). Under certain assumptions on \(\mathscr {L}\), we prove for any given boundary data the existence of a map which is:

1. (i)
   a vectorial Absolute Minimiser of (1) in the sense of Aronsson,
    
2. (ii)
   a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations,
    
3. (iii)
   a limit of minimisers of the respective \(L^p\) functionals as \(p\rightarrow \infty \) for any \(q\ge 1\) in the strong \(W^{1,q}\) topology and
    
4. (iv)
   partially \(C^2\) on \(\Omega \) off an exceptional compact nowhere dense set.
    

Our method is based on \(L^p\) approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of \(\mathcal {D}\)-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.

     

Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects«

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise

1. (1)
   God is possible.
    

Second, a demonstration of the “remarkable proposition”

1. (2)
   If God is possible, then God exists.
    

The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows:

1. (3)
   \(g=_{\mathrm{df}} \,\upiota x\square E(x)\).
    

Proposition (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:

1. (4)
   \(\diamondsuit E(\upiota x\square E(x)) \rightarrow \, E(\upiota x\square E(x))\).
    

At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \(\hbox {F}(y)\) and \(\lnot F(y)\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of (2) turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of (1), i.e. the self-consistency of the concept of a necessary being, remains to be proven.

     

The inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?
    

     

Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules

In this paper, we investigate compactly supported symmetric orthonormal dyadic complex wavelets such that the symmetric orthonormal refinable functions have high linear-phase moments and the antisymmetric wavelets have high vanishing moments. Such wavelets naturally lead to real-valued symmetric tight wavelet frames with some desirable moment properties, and are related to coiflets which are real-valued and are of interest in numerical algorithms. For any positive integer m, employing only the Riesz lemma without solving any nonlinear equations, we obtain a 2π-periodic trigonometric polynomial \(\hat a\) with complex coefficients such that

1. (i)
   \(\hat a\) is an orthogonal mask: \(|\hat a(\xi)|^2+|\hat a(\xi+\pi)|^2=1\).
    
2. (ii)
   \(\hat a\) has m?+?1???odd _m sum rules: \(\hat a(\xi+\pi)=O(|\xi|^{m+1-odd_m})\) as ξ→0, where \(odd_m:=\frac{1-(-1)^m}{2}\).
    
3. (iii)
   \(\hat a\) has m?+?odd _m linear-phase moments: \(\hat a(\xi)=e^{{{\mathrm{i}}} c\xi}+O(|\xi|^{m+odd_m})\) as ξ→0 with phase c?=???1/2.
    
4. (iv)
   \(\hat a\) has symmetry and coefficient support [2???2m,2m???1]: \(\hat a(\xi)=\sum_{k=2-2m}^{2m-1} h_k e^{-{{\mathrm{i}}} k\xi}\) with h_1???k ?=?h _k .
    
5. (v)
   \(\hat a(\xi)\ne 0\) for all ξ?∈?(???π,π).
    

Define \(\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi)\) and \(\hat \psi(2\xi)=e^{-{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi)\). Then ψ is a compactly supported antisymmetric orthonormal wavelet with m?+?1???odd _m vanishing moments, and ? is a compactly supported symmetric orthonormal refinable function with the special linear-phase moments: \(\int_{{{\mathbb R}}} \phi(x)dx=1\) and \(\int_{{{\mathbb R}}} (x-1/2)^j \phi(x) dx=0\) for all j?=?1,...,m?+?odd _m ???1. Both functions ? and ψ are supported on [2???2m,2m???1].The mask of a coiflet has real coefficients and satisfies (i), (ii), and (iii), often with a general phase c and the additional condition that the order of the linear-phase moments is equal (or close) to the order of the sum rules. On the one hand, as Daubechies showed in [3, 5] that except the Haar wavelet, any compactly supported dyadic orthonormal real-valued wavelets including coiflets cannot have symmetry. On the other hand, solving nonlinear equations, [4, 12] constructed many interesting real-valued dyadic coiflets without symmetry. But it remains open whether there is a family of real-valued orthonormal wavelets such as coiflets whose masks can have arbitrarily high linear-phase moments. This partially motivates this paper to study the complex wavelet case with symmetry property. Though symmetry can be achieved by considering complex wavelets, the symmetric Daubechies complex orthogonal masks in [11] generally have no more than 2 linear-phase moments. In this paper, we shall study and construct orthonormal dyadic complex wavelets and masks with symmetry, linear-phase moments, and sum rules. Examples and two general construction procedures for symmetric orthogonal masks with high linear-phase moments and sum rules are given to illustrate the results in this paper. We also answer an open question on construction of symmetric Daubechies complex orthogonal masks in the literature.     

The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if \(\mu \) is a Borel measure in \({{\mathbb {R}}}^{n+1}\) with growth of order n, such that the n-dimensional Riesz transform \({{\mathcal {R}}}_\mu \) is bounded in \(L^2(\mu )\), and \(B\subset {{\mathbb {R}}}^{n+1}\) is a ball with \(\mu (B)\approx r(B)^n\) such that:

1. (a)
   there is some n-plane L passing through the center of B such that for some \(\delta >0\) small enough, it holds

   $$\begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned}$$
    
2. (b)
   for some constant \({\varepsilon }>0\) small enough,

   $$\begin{aligned}\int _{B} |{{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1)|^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned}$$

   where \(m_{\mu ,B}({{\mathcal {R}}}_\mu 1)\) stands for the mean of \({{\mathcal {R}}}_\mu 1\) on B with respect to \(\mu \),
    

then there exists a uniformly n-rectifiable set \(\Gamma \), with \(\mu (\Gamma \cap B)\gtrsim \mu (B)\), and such that \(\mu |_\Gamma \) is absolutely continuous with respect to \({{\mathcal {H}}}^n|_\Gamma \). This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.

     

Flow equivalence of sofic shifts

We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients.

1. (1)
   An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them.
    
2. (2)
   The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs.
    

     

Limit problems for a Fractional p-La placian as $${ p\to\infty}$$

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),

$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$

where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases:

1. (i)
   \({f=f(x).}\)
    
2. (ii)
   \({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
    

We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:

$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$

where

$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$

     

Korn’s inequality and John domains

Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent:

1. (i)
   \(\Omega \) is a John domain;
    
2. (ii)
   for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),

   $$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$
    
3. (ii’)
   for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
    
4. (iii)
   for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with

   $$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$
    
5. (iii’)
   for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
    

For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.

     

<Emphasis Type="Italic">b</Emphasis>-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) an b-generalized skew derivation of R, L a non-central Lie ideal of R, \(0\ne a\in R\) and \(n\ge 1\) a fixed integer. In this paper, we prove the following two results:

1. 1.
   If R has characteristic different from 2 and 3 and \(a[F(x),x]^n=0\), for all \(x\in L\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity of degree 4, and there exist \(\lambda \in C\) and \(b\in Q\), such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\).
    
2. 2.
   If \(\mathrm{{char}}(R)=0\) or \(\mathrm{{char}}(R) > n\) and \(a[F(x),x]^n\in Z(R)\), for all \(x\in R\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\).
    

     

More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).     

Multi-invariant measures and subsets on nilmanifolds

Given a Z^r-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action

1. (1)
   the factor measure has zero entropy under every element of the action
    
2. (2)
   the factor action is virtually cyclic.
    

We also deduce a rigidity property for invariant closed subsets.

     

Coloring of the Square of Kneser Graph $$$$

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n elements set, with two vertices adjacent if they are disjoint. The square \(G^2\) of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in \(G^2\) if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(n, k) is an interesting graph coloring problem, and is also related with intersecting family problem. The square of K(2k, k) is a perfect matching and the square of K(n, k) is the complete graph when \(n \ge 3k-1\). Hence coloring of the square of \(K(2k +1, k)\) has been studied as the first nontrivial case. In this paper, we focus on the question of determining \(\chi (K^2(2k+r,k))\) for \(r \ge 2\). Recently, Kim and Park (Discrete Math 315:69–74, 2014) showed that \(\chi (K^2(2k+1,k)) \le 2k+2\) if \( 2k +1 = 2^t -1\) for some positive integer t. In this paper, we generalize the result by showing that for any integer r with \(1 \le r \le k -2\),

1. (a)
   \(\chi (K^2 (2k+r, k)) \le (2k+r)^r\),   if   \(2k + r = 2^t\) for some integer t, and
    
2. (b)
   \(\chi (K^2 (2k+r, k)) \le (2k+r+1)^r\),   if  \(2k + r = 2^t-1\) for some integer t.
    

On the other hand, it was shown in Kim and Park (Discrete Math 315:69–74, 2014) that \(\chi (K^2 (2k+r, k)) \le (r+2)(3k + \frac{3r+3}{2})^r\) for \(2 \le r \le k-2\). We improve these bounds by showing that for any integer r with \(2 \le r \le k -2\), we have \(\chi (K^2 (2k+r, k)) \le 2 \left( \frac{9}{4}k + \frac{9(r+3)}{8} \right) ^r\). Our approach is also related with injective coloring and coloring of Johnson graph.

     

Chvátal–Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs

For a graph H, let \(\alpha (H)\) and \(\alpha ^{\prime }(H)\) denote the independence number and the matching number, respectively. Let \(k\ge 2\) and \(r>0\) be given integers. We prove that if H is a k-connected claw-free graph with \(\alpha (H)\le r\), then either H is Hamiltonian or the Ryjá c? ek’s closure \(cl(H)=L(G)\) where G can be contracted to a k-edge-connected \(K_3\)-free graph \(G_0^{\prime }\) with \(\alpha ^{\prime }(G_0^{\prime })\le r\) and \(|V(G_0^{\prime })|\le \max \{3r-5, 2r+1\}\) if \(k\ge 3\) or \(|V(G_0^{\prime })|\le \max \{4r-5, 2r+1\}\) if \(k=2\) and \(G_0^{\prime }\) does not have a dominating closed trail containing all the vertices that are obtained by contracting nontrivial subgraphs. As corollaries, we prove the following:

1. (a)
   A 2-connected claw-free graph H with \(\alpha (H)\le 3\) is either Hamiltonian or \(cl(H)=L(G)\) where G is obtained from \(K_{2,3}\) by adding at least one pendant edge on each degree 2 vertex;
    
2. (b)
   A 3-connected claw-free graph H with \(\alpha (H)\le 7\) is either Hamiltonian or \(cl(H)=L(G)\) where G is a graph with \(\alpha ^{\prime }(G)=7\) that is obtained from the Petersen graph P by adding some pendant edges or subdividing some edges of P.
    

Case (a) was first proved by Xu et al. [19]. Case (b) is an improvement of a result proved by Flandrin and Li [12]. For a given integer \(r>0\), the number of graphs of order at most \(\max \{4r-5, 2r+1\}\) is fixed. The main result implies that improvements to case (a) or (b) by increasing the value of r and by enlarging the collection of exceptional graphs can be obtained with the help of a computer. Similar results involved degree or neighborhood conditions are also discussed.