A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Properties of minimally t-tough graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta \left(G\right)=2$. We show that in every minimally $1$-tough graph $\delta \left(G\right)\le \frac{n}{3}+1$. We also prove that every minimally $1$-tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number $t$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.     

Subspace partitions of Fqn containing direct sums

Let $q$ be a prime power and $n$ be a positive integer. A subspace partition of $V={\mathbb{F}}_q^n$, the vector space of dimension $n$ over ${\mathbb{F}}_q$, is a collection $\Pi$ of subspaces of $V$ such that each nonzero vector of $V$ is contained in exactly one subspace in $\Pi$; the multiset of dimensions of subspaces in $\Pi$ is then called a Gaussian partition of $V$. We say that $\Pi$contains a direct sum if there exist subspaces $W_1,\dots ,W_k\in \Pi$ such that $W_1\oplus ?\oplus W_k=V$. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers $a_1$ and $a_2$ with $n>a_1>a_2\ge 1$, our main theorem shows that if $\Pi$ is a subspace partition of ${\mathbb{F}}_q^n$ with $m_i$ subspaces of dimension $a_i$ for $i=1,2$, then $\Pi$ contains a direct sum when $a_1{x}_1+a_2{x}_2=n$ has a solution $(x_1,x_2)$ for some integers $x_1,x_2\ge 0$ and $m_2$ belongs to the union $I$ of two natural intervals. The lower bound of $I$ captures all subspace partitions with dimensions in $\{a_1,a_2\}$ that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when $m_2?I$ or when the condition on the existence of a nonnegative integral solution $(x_1,x_2)$ is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when $q\to 1$) as well as subspace and set partitions.     

A degree condition for a graph to have (a,b)-parity factors

Let $a$ and $b$ be two positive integers such that $a\le b$ and $a\equiv b(mod2)$. A graph $F$ is an $(a,b)$-parity factor of a graph $G$ if $F$ is a spanning subgraph of $G$ and for all vertices $v\in V\left(F\right)$, $d_F\left(v\right)\equiv b(mod2)$ and $a\le d_F\left(v\right)\le b$. In this paper we prove that every connected graph $G$ with $n\ge b(a+b)(a+b+2)∕\left(2a\right)$ vertices has an $(a,b)$-parity factor if $na$ is even, $\delta \left(G\right)\ge (b?a)∕a+a$, and for any two nonadjacent vertices $u,v\in V\left(G\right)$, $max\{d_G\left(u\right),d_G\left(v\right)\}\ge \frac{an}{a+b}$. This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998).     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

d-matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A $d$-matching in a 3-uniform hypergraph $H$ is a matching of size $d$. Let $V_1,V_2$ be a partition of $n$ vertices such that $\left|V_1\right|=2d?1$ and $\left|V_2\right|=n?2d+1$. Denote by $E_3(2d?1,n?2d+1)$ the 3-uniform hypergraph with vertex set $V_1\cup V_2$ consisting of all those edges which contain at least two vertices of $V_1$. Let $H$ be a 3-uniform hypergraph of order $n\ge 9d^2$ such that $deg\left(u\right)+deg\left(v\right)>2\left[\left(\genfrac{}{}{0ex}{}{n?1}{2}\right)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)\right]$ for any two adjacent vertices $u,v\in V\left(H\right)$. In this paper, we prove $H$ contains a $d$-matching if and only if $H$ is not a subgraph of $E_3(2d?1,n?2d+1)$.     

Moments about the mean of the size of a self-conjugate (s,t)-core partition

Johnson proved that if $s,t$ are coprime integers, then the $r$th moment of the size of an $(s,t)$-core is a polynomial of degree $2r$ in $t$ for fixed $s$. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of ${\stackrel{?}{\mathfrak{S}}}_t∕{\mathfrak{S}}_t$ and $t$-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an $(s,t)$-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an $(s,t)$-core and the first nine moments about the mean of the size of an $(s,s+1)$-core using Maple. To get the moments about the mean of the size of a self-conjugate $(s,t)$-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if $s,t$ are coprime integers, then the $r$th moment about the mean of the size of a self-conjugate $(s,t)$-core is a quasipolynomial of period 2 and degree $2r$ in $t$ for fixed odd $s$. Then, based on a bijection of Ford, Mai and Sze between self-conjugate $(s,t)$-cores and lattice paths in $?\frac{s}{2}?\times ?\frac{t}{2}?$ rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate $(s,t)$-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate $(s,t)$-core.     

The list L(2,1)-labeling of planar graphs

Let $\mathbf{N}$ be the set of all positive integers. A list assignment of a graph $G$ is a function $L:V\left(G\right)?2^{\mathbf{N}}$ that assigns each vertex $v$ a list $L\left(v\right)$ for all $v\in V\left(G\right)$. We say that $G$ is $L$-$(2,1)$-choosable if there exists a function $?$ such that $?\left(v\right)\in L\left(v\right)$ for all $v\in V\left(G\right)$, $|?\left(u\right)??\left(v\right)|\ge 2$ if $u$ and $v$ are adjacent, and $|?\left(u\right)??\left(v\right)|\ge 1$ if $u$ and $v$ are at distance 2. The list-$L(2,1)$-labeling number ${\lambda }_l\left(G\right)$ of $G$ is the minimum $k$ such that for every list assignment $L=\{L\left(v\right):|L\left(v\right)|=k,v\in V\left(G\right)\}$, $G$ is $L$-$(2,1)$-choosable. We prove that if $G$ is a planar graph with girth $g\ge 8$ and its maximum degree $\Delta$ is large enough, then ${\lambda }_l\left(G\right)\le \Delta +3$. There are graphs with large enough $\Delta$ and $g\ge 8$ having ${\lambda }_l\left(G\right)=\Delta +3$.     

On the spanning connectivity of the generalized Petersen graphs P(n,3)

In this paper, we employed lattice model to describe the three internally vertex-disjoint paths that span the vertex set of the generalized Petersen graph $P(n,3)$. We showed that the $P(n,3)$ is 3-spanning connected for odd $n$. Based on the lattice model, five amalgamated and one extension mechanisms are introduced to recursively establish the 3-spanning connectivity of the $P(n,3)$. In each amalgamated mechanism, a particular lattice trail was amalgamated with the lattice trails that was dismembered, transferred, or extended from parts of the lattice trails for $P(n?6,3)$, where a lattice tail is a trail in the lattice model that represents a path in $P(n,3)$.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

2p-cycle decompositions of some regular graphs and digraphs

In this paper, we consider $2k$-cycle decomposition of $K_m\times K_n$ and directed $2k$-cycle decompositions of ${(K_m°{\overline{K}}_n)}^1$ and ${(K_m\times K_n)}^1$, where $°$ and $\times$ denote the wreath product and tensor product of graphs, respectively. Using the results obtained here, we prove that for $m,n\ge 3$, the obvious necessary conditions for the existence of a $C_{2k}$-decomposition of $K_m\times K_n$ are sufficient whenever $k\in \{p,2^{?}\},$ where $p$ is a prime and $?\ge 2$. Also, we show that the necessary conditions for the existence of ${\overrightarrow{C}}_{2p}$-decompositions of ${(K_m°{\overline{K}}_n)}^1$ and ${(K_m\times K_n)}^1$ are sufficient whenever $p$ is a prime, where ${\overrightarrow{C}}_{2p}$ denotes the directed cycle of length $2p$.     

A pairwise almost disjoint subspace of kN of maximal dimension

We show that if k is an infinite field, then there exists a subspace $W?k^{\mathbb{N}}$ of dimension $|k{|}^{{\mathrm{?}}_0}$, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.