The genus of the binomial random graph $G(n,p)$ is well understood for a wide range $p=p(n)$. Recently, study bipartite $G(n_1,n_2,p)$, with partition classes size $n_1$ and $n_2$, was initiated by Mohar Jing, who showed that when $n_2$ are comparable in $p=p(n_1,n_2)$ significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ has similar behavior to graph. In this paper we show there threshold planar...

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical problem is to minimize the size of a set S vertices an n-vertex graph G such that $$G - N[S]$$ , by deleting closed neighborhood S, contains no vertices. In proof Art Gallery Theorem, Chvátal showed minimum size, called number and denoted $$\gamma (G)$$ at most n/3 if mop. Here we consider modific...

We study the problem of Embedding Wirelength $n$-dimensional Hypercube $Q_n$ into Cylinder $C_{2^{n_1}}\times P_{2^{n_2}}$, where $n_1+ n_2=n$, called EWHC. show that such wirelength corresponding to Gray code embedding is $2^{n_2}(3\cdot 2^{2n_1-3}-2^{n_1-1})+2^{n_1} (2^{2n_2-1}-2^{n_2-1})$. In addition, we prove an optimal strategy

Let $R$ be a prime ring, $I$ be any nonzero ideal of $R$ and $f:Irightarrow R$ be an additivemap. Then skew-Engel condition $langle... langle langle$$f(x),x^{n_1} rangle,x^{n_2} rangle ,...,x^{n_k} rangle=0$ implies that $f (x)=0$ $forall,xin I$ provided $2neq$ char $(R)>n_1+n_2+...+n_k, $ where $n_1,n_2,...,n_k$ are natural numbers. This extends some existing results. In the end, we also gener...

For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, $G+e$. The saturation number $F$, denoted $sat(F,H)$, the minimum edges an subgraph $F$. In this paper, we study numbers $tK_{l,l,l}$ complete tripartite $K_{n_1,n_2,n_3}$. $t\ge 1$, $l\ge 1$ $n_1,n_2$ $n_3$ sufficiently large, determine $sat(K_{n_1,n...

Abstract Let ${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m -edge subgraph of the complete bipartite graph ${K_{n_1,n_2}}$ with bipartition $(V_1, V_2)$ , where $n_i = |V_i|$ $i=1,2$ . Given real number $p \in [0,1]$ such that $d_1 \,{:\!=}\, pn_2$ and $d_2 pn_1$ are integers, let $\mathbb{R}(n_1,n_2,p)$ every vertex $v V_i$ degree $d_i$ $i 1, 2$ In this paper we determine sufficient conditi...

Let \(P(G, x)\) be a chromatic polynomial of graph \(G\). Two graphs \(G\) and \(H\) are called chromatically equivalent iff x) = H(G, x)\). A is unique if \(G\simeq H\) for every to In this paper, the uniqueness complete tripartite \(K(n_1, n_2, n_3)\) proved \(n_1 \geqslant n_2 n_3 2\) - \leqslant 5\).