On Bipolar Anti Fuzzy h-ideals in Hemi-rings

In this work, we present notions of bipolar anti fuzzy $h$-ideals and bipolar anti fuzzy interior $h$-ideals in hemi-rings. Investigating some of their properties, we characterize hemi-rings by means of positive anti $\beta$-cut and negative anti $\alpha$-cut. Meanwhile, some results of homomorphisms, anti images and anti pre-images are given to show the rationality of the definitions introduced in the present paper. Also, we define an equivalence relation on bipolar anti fuzzy $h$-ideals. In particular, we investigate translations, extensions and multiplications of bipolar anti fuzzy $h$-ideals. Finally, we present characterizations of h-hemi-regular and h-semi-simple hemi-rings in terms of bipolar anti fuzzy $h$-ideals.     

Recolouring reflexive digraphs

Given digraphs $G$ and $H$, the colouring graph $\mathbf{Col}(G,H)$ has as its vertices all homomorphism of $G$ to $H$. There is an arc $?\to {?}^{\prime }$ between two homomorphisms if they differ on exactly one vertex $v$, and if $v$ has a loop we also require $?\left(v\right)\to {?}^{\prime }\left(v\right)$. The recolouring problem asks if there is a path in $\mathbf{Col}(G,H)$ between given homomorphisms $?$ and $\psi$. We examine this problem in the case where $G$ is a digraph and $H$ is a reflexive, digraph cycle.We show that for a reflexive digraph cycle $H$ and a reflexive digraph $G$, the problem of determining whether there is a path between two maps in $\mathbf{Col}(G,H)$ can be solved in time polynomial in $G$. When $G$ is not reflexive, we show the same except for certain digraph 4-cycles $H$.     

A new class of n-ary hyperoperations

To a given $n$-ary hyperoperation $f$ on a universe $H$ and a unary hyperoperation $\phi$ on $H$ we define a new $n$-ary hyperoperation ${\partial }_f^{\phi }$ on $H$. We study the associativity, weak associativity and reproductivity of $n$-ary hyperoperations ${\partial }_f^{\phi }$.     

The size of a graph is reconstructible from any n−2 cards

Let $G$ and $H$ be graphs of order $n$. The number of common cards of $G$ and $H$ is the maximum number of disjoint pairs $(v,w)$, where $v$ and $w$ are vertices of $G$ and $H$, respectively, such that $G?v?H?w$. We prove that if the number of common cards of $G$ and $H$ is at least $n?2$ then $G$ and $H$ must have the same number of edges when $n\ge 29$. This is the first improvement on the $25$-year-old result of Myrvold that if $G$ and $H$ have at least $n?1$ common cards then they have the same number of edges. It also improves on the result of Woodall and others that the numbers of edges of $G$ and $H$ differ by at most $1$ when they have $n?2$ common cards.     

On left (right) ρ-transitive geometric spaces

In this article, first we generalize the concept of $\mathfrak{B}$-parts in a geometric space with respect to a binary relation $\rho$ and then we study left (right) $l{\mathfrak{B}}_{\rho }$-closures ($r{\mathfrak{B}}_{\rho }$-closures, respectively). Finally we introduce and investigate the notions left $\rho$-strongly transitive geometric spaces and left $\rho$-emphatic strongly transitive geometric spaces.     

Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths

It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix ($H$-cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to $P_n$ is $H$-cospectral to $P_n$. Interestingly, there are others. This paper finds digraphs that are $H$-cospectral with the path graph $P_n$ and whose underlying graphs are nonisomorphic, when $n$ is odd, and finds that such graphs do not exist when $n$ is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined.     

Self-orthogonal decompositions of graphs into matchings

Given a simple graph H, a self-orthogonal decomposition (SOD) of H is a collection of subgraphs of H, all isomorphic to some graph G, such that every edge of H occurs in exactly two of the subgraphs and any two of the subgraphs share exactly one edge. Our concept of SOD is a natural generalization of the well-studied orthogonal double covers (ODC) of complete graphs. If for some given G there is an appropriate H, then our goal is to find one with as few vertices as possible. Special attention is paid to the case when G a matching with $n-1$ edges. We conjecture that $v(H)=2n-2$ is best possible if $n\ne 4$ is even and $v(H)=2n$ if n is odd. We present a construction which proves this conjecture for all but 4 of the possible residue classes of n modulo 18.     

The edge spectrum of the saturation number for small paths

Let $H$ be a simple graph. A graph $G$ is called an $H$-saturated graph if $H$ is not a subgraph of $G$, but adding any missing edge to $G$ will produce a copy of $H$. Denote by $SAT(n,H)$ the set of all $H$-saturated graphs $G$ with order $n$. Then the saturation number $sat(n,H)$ is defined as ${min}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$, and the extremal number $ex(n,H)$ is defined as ${max}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$. A natural question is that of whether we can find an $H$-saturated graph with $m$ edges for any $sat(n,H)\le m\le ex(n,H)$. The set of all possible values $m$ is called the edge spectrum for $H$-saturated graphs. In this paper we investigate the edge spectrum for $P_i$-saturated graphs, where $2\le i\le 6$. It is trivial for the case of $P_2$ that the saturated graph must be an empty graph.     

On the lower central series of the IA-automorphism group of a free group

In this paper, we study the Johnson homomorphisms ${\tau }_k^{\prime }$ of the automorphism group of a free group of rank $n$, which are defined on the graded quotients of the lower central series of the IA-automorphism group. In particular, we determine the cokernel of ${\tau }_k^{\prime }$ for any $k\ge 2$ and $n\ge k+2$.     

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 $t?1$.