We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let $D=(V,A)$ be a reflexive digraph (or network). Consider $X$ and $Y$ (not necessarily disjoint) nonempty subsets of vertices (or nodes) of $D$. A disimplex $K(X,Y)$ of $D$ is...

A majority coloring of a digraph $D$ with $k$ colors is an assignment $\pi :V\left(D\right)\to \{1,2,\cdots ,k\}$ such that for every $v\in V\left(D\right)$ we have $\pi \left(w\right)=\pi \left(v\right)$ for at most half of all out-neighbors $w\in N^{+}\left(v\right)$. A digraph $D$ is majority $k$-choosable if for any assignment of lists of colors of size $k$ to the vertices, there is a majority coloring of $D$ from these lists. We prove that if $U\left(D\right)$ is a 1-planar graph without a 4-cycle, then $D$ is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable.

For a digraph $D$, the niche hypergraph $N\mathscr{H}\left(D\right)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges $E\left(N\mathscr{H}\right(D\left)\right)=\{e\subseteq V(D):|e|\ge 2$ and there exists a vertex $v$ such that $e=N_D^{-}\left(v\right)$ or $e=N_D^{+}\left(v\right)\}$. A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $\mathscr{H}$, the niche number $\widehat{n}\left(\mathscr{H}\right)$ is the smallest integer such that $\mathscr{H}$ together with $\widehat{n}\left(\mathscr{H}\right)$ isolated vertices is the niche hypergraph of an acyclic digraph. C. Garske, M. Sonntag and H. M. Teichert (2016) conjectured that for a linear...

For any two positive integers $n$ and $k\ge 2$, let $G(n,k)$ be a digraph whose set of vertices is $\{0,1,...,n-1\}$ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k\equiv b(modn)$. Let $n={\prod }_{i=1}^r{p}_i^{e_i}$ be the prime factorization of $n$. Let $P$ be the set of all primes dividing $n$ and let $P_1,P_2\subseteq P$ be such that $P_1\cup P_2=P$ and $P_1\cap P_2=\varnothing$. A fundamental constituent of $G(n,k)$, denoted by $G_{P_2}^{*}(n,k)$, is a subdigraph of $G(n,k)$ induced on the set of vertices which are multiples of ${\prod }_{p_i\in P_2}{p}_i$ and are relatively prime to all primes $q\in P_1$. L. Somer and M. Křížek proved that the trees attached...

An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on $n$ vertices with minimum outdegree $d$ contains a directed cycle of length at most $\lceil n/d\rceil$. In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that ${\alpha }_0$ is the smallest real such that every $n$-vertex digraph with minimum outdegree at least ${\alpha }_{0}n$ contains a directed triangle. Let $ϵ<(3-2{\alpha }_0)/(4-2{\alpha }_0)$ be a positive real. We show that if $D$ is an oriented graph...

Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If ${\sum }_{x\in N¯\left[v\right]}f\left(x\right)\ge 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number ${\gamma }_S\left(D\right)$ of D. A set $f₁,f₂,...,f_d$ of signed dominating functions on D with the property that...

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

Let $G$ be a finite group and let $\pi \left(G\right)=\{p_1,p_2,...,p_k\}$ be the set of prime divisors of $\left|G\right|$ for which $p_1<p_2<\cdots <p_k$. The Gruenberg-Kegel graph of $G$, denoted $GK\left(G\right)$, is defined as follows: its vertex set is $\pi \left(G\right)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i{p}_j$. The degree of a vertex $p_i$ in $\mathrm{GK}\left(G\right)$ is denoted by $d_G\left(p_i\right)$ and the $k$-tuple $D\left(G\right)=(d_G\left(p_1\right),d_G\left(p_2\right),...,d_G\left(p_k\right))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi \left(G\right)$ is the vertex set of a connected component of $GK\left(G\right)$, then the largest $\omega$-number which divides $\left|G\right|$, is...

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.