Homomorphism complexes and k-cores

For any fixed graph $G$, we prove that the topological connectivity of the graph homomorphism complex Hom($G,K_m$) is at least $m?D\left(G\right)?2$, where $D\left(G\right)={max}_{H?G}\delta \left(H\right)$, for $\delta \left(H\right)$ the minimum degree of a vertex in a subgraph $H$. This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree $\Delta \left(G\right)$ was used in place of $D\left(G\right)$, and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, $\chi \left(G\right)\le D\left(G\right)+1$, as $\chi \left(G\right)=min\{m:\text{Hom}(G,K_m)\ne ?\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c∕n$ for a fixed constant $c>0$.     

Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.     

Mappings preserving B-orthogonality

It is well known that a linear mapping $T:X\to Y$ preserving the Birkhoff orthogonality (i.e. ${?}_{x,y\in X}x{\perp }_{\text{B}}y?Tx{\perp }_{\text{B}}Ty$), has to be a similarity. For real spaces it has been proved by Koldobsky (1993); a proof including both real and complex spaces has been given by Blanco and Turn?ek (2006). In the present paper the author would like to present a somewhat simpler proof of this nice theorem. Moreover, we extend the Koldobsky theorem; more precisely, we show that the linearity assumption may be replaced by additivity.     

Combinatorics of n-color cyclic compositions

Integer compositions and related enumeration problems have been of interest to combinatorialists and number theorists for a long time. The cyclic and colored analogues of this concept, although interesting, have not been extensively studied. In this paper we explore the combinatorics of $n$-color cyclic compositions, presenting generating functions, bijections, asymptotic formulas related to the number of such compositions, the number of parts, and the number of restricted parts.     

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.     

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 $E_3(2d?1,n?2d+1)$.     

On q-Steiner systems from rank metric codes

In this paper we prove that rank metric codes with special properties imply the existence of $q$-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a $[2d,d,d]$ dually almost MRD code $C\le {\mathbb{F}}_{q^m}^{2d}$$(2d\le m)$ which has no code words of rank weight $d+1$ form a $q$-Steiner system $S{(d?1,d,2d)}_q$. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.     

On Lie p-algebras of cohomological dimension one

We prove that a Lie $p$-algebra of cohomological dimension one is one-dimensional, and discuss related questions.     

On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).     

On images of locally finite derivations and E-derivations

Some cases of the LFED Conjecture, proposed by the second author [15], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{?1}]$, where $x=(x_1,x_2,\dots ,x_n)$ denotes n commutative free variables and k a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat–van der Kallen Theorem [3] is also discussed and emphasized.     

The unique solution for a fractional q-difference equation with three-point boundary conditions

The aim of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional $q$-difference equations with three-point boundary conditions. Our approach relies on a new fixed point theorem of increasing $\psi ?(h,r)?$concave operators defined on ordered sets. Further, we can construct a monotone explicit iterative scheme to approximate the unique solution. Finally, the main results are illustrated with the aid of two interesting examples.     

List r-hued chromatic number of graphs with bounded maximum average degrees

For integers $k,r>0$, a $(k,r)$-coloring of a graph $G$ is a proper coloring $c$ with at most $k$ colors such that for any vertex $v$ with degree $d\left(v\right)$, there are at least min$\{d\left(v\right),r\}$ different colors present at the neighborhood of $v$. The $r$-hued chromatic number of $G$, ${\chi }_r\left(G\right)$, is the least integer $k$ such that a $(k,r)$-coloring of $G$ exists. The list$r$-hued chromatic number${\chi }_{L,r}\left(G\right)$ of $G$ is similarly defined. Thus if $\Delta \left(G\right)\ge r$, then ${\chi }_{L,r}\left(G\right)\ge {\chi }_r\left(G\right)\ge r+1$. We present examples to show that, for any sufficiently large integer $r$, there exist graphs with maximum average degree less than 3 that cannot be $(r+1,r)$-colored. We prove that, for any fraction $q<\frac{14}{5}$, there exists an integer $R=R\left(q\right)$ such that for each $r\ge R$, every graph $G$ with maximum average degree $q$ is list $(r+1,r)$-colorable. We present examples to show that for some $r$ there exist graphs with maximum average degree less than 4 that cannot be $r$-hued colored with less than $\frac{3r}{2}$ colors. We prove that, for any sufficiently small real number $?>0$, there exists an integer $h=h\left(?\right)$ such that every graph $G$ with maximum average degree $4??$ satisfies ${\chi }_{L,r}\left(G\right)\le r+h\left(?\right)$. These results extend former results in Bonamy et al. (2014).     

An example showing that A-lower semi-continuity is essential for minimax continuity theorems

Recently the authors have established continuity properties of minimax values and solution sets for a function of two variables depending on a parameter. Some of these properties hold under the assumption that the multifunction, defining the domains of the second variable, is $A$-lower semi-continuous. This property is stronger than lower semi-continuity, but in several important cases these two properties coincide. This note provides an example demonstrating that in general the $A$-lower semi-continuity assumption cannot be relaxed to lower semi-continuity.     

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.     

Characterization of the induced matching extendable graphs with 2n vertices and 3n edges

A graph $G$ is induced matching extendable or IM-extendable if every induced matching of $G$ is contained in a perfect matching of $G$. In 1998, Yuan proved that a connected IM-extendable graph on $2n$ vertices has at least $3n?2$ edges, and that the only IM-extendable graph with $2n$ vertices and $3n?2$ edges is $T\times K_2$ , where $T$ is an arbitrary tree on $n$ vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with $2n\ge 6$ vertices and $3n?1$ edges is $T\times K_2+e$, where $T$ is an arbitrary tree on $n$ vertices and $e$ is an edge connecting two vertices that lie in different copies of $T$ and have distance 3 between them in $T\times K_2$. In this paper, we introduced the definition of $Q$-joint graph and characterized the connected IM-extendable graphs with $2n\ge 4$ vertices and $3n$ edges.