For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of vertices there exists monochromatic $$a_i$$ -clique in some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free ,a_r)^v\}$$ where H graph. Such have been extensively studied $$H=K_s$$ with...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.

For a graph H $H$ , let c ( ) = inf { : e G ≥ | implies ≻ } $c(H)=\text{inf}\{c:e(G)\ge c|G|\,\,\text{implies}\,\,G\succ H\}$ where $G\succ H$ means that is minor of $G$ . We show if has average degree d $d$ then ≤ 0.319 … + o 1 log $c(H)\le (0.319\,\ldots \,+{o}_{d}(1))|H|\sqrt{\mathrm{log}d}$ $0.319\ldots $ an explicitly defined constant. This bound matches corresponding lower shown to hold f...

Abstract In this paper we study the asymptotic behaviour of a random uniform parking function $\pi_n$ size n . We show that first $k_n$ places $\pi_n(1),\ldots,\pi_n(k_n)$ are asymptotically independent and identically distributed (i.i.d.) on $\{1,2,\ldots,n\}$ , for total variation distance when $k_n = {\rm{o}}(\sqrt{n})$ Kolmogorov $k_n={\rm{o}}(n)$ improving results Diaconis Hicks. Moreover,...

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on influence a parameter $a$ which controls strength advection. For infinite domain we find new critical value $a_c=0.6890665337007457\ldots$ below there % if write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit i...

Given a number field $K$, finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct extensions $L/K$ with Galois that realise all of the $\alpha_1,\ldots,\alpha_t$ as norms in $L$. In particular, this shows existence such for any given parameters. Our approach relies on class theory recent formulation Tate's characterisation Hasse norm principle, local-...

Let $G$ be a simple graph on $n$ vertices. $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal parity binomial edge of in polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] respectively. We classify graphs whose LSS ideals are complete intersections. also almost intersections, we prove that their Rees algebra is Cohen-Macaulay. compute second grade...

Abstract We study membership of rational inner functions in Dirichlet-type spaces polydiscs. In particular, we prove a theorem relating such inclusions to $$H^p$$ H p integrability partial derivatives an RIF, and as corollary that all on $${\mathbb {D}}^n$$ <...

Let $$\xi _1,\xi _2,\ldots $$ be a sequence of independent copies random vector in $$\mathbb {R}^d$$ having an absolutely continuous distribution. Consider walk $$S_i:=\xi _1+\cdots +\xi _i$$ , and let $$C_{n,d}:={{\,\mathrm{conv}\,}}(0,S_1,S_2,\ldots ,S_n)$$ the convex hull first $$n+1$$ points it has visited. The polytope $$C_{n,d}$$ is called k-neighborly if for any indices $$0\le i_1<\cdots...

Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ characteristic zero, $L_{n,c}=L_{n}/(L_{n}''+\gamma_{c+1}(L_{n}))$ metabelian nilpotent class $c$ algebra, and $F_{n}=L_{n}/L_{n}''$ generated by $x_1,\ldots,x_n$ zero. We call polynomial $p(X_n)$ in these algebras {\it symmetric} if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element symmetric group $S_n$. The se...