Motivated by an optimal visiting problem, we study a switching mean-field game on network, where both decisional and time-variable is at disposal of the agents for what concerns, respectively, instant to decide perform switch. Every switch between nodes network represents from $0$ $1$ one component string $p = (p_1,\ldots, p_n)$ which, in interpretation, gives information visited targets, being...

Let \( \mathcal B= (B_1,\ldots , B_h)\) be an h-tuple of sets positive integers. \(g_{ B}(n)\) count the number representations n in form \(n = b_1\cdots b_h\), where \(b_i \in B_i\) for all \(i \{1,\ldots h\}\). It is proved that \(\liminf _{n\rightarrow \infty } g_{ B}(n) \ge 2\) implies \(\limsup \).

Consider the following game between Builder and Painter. We take some families of graphs $\mathcal{G}_{1},\ldots,\mathcal{G}_t$ an integer $n$ such that $n \geq R(\mathcal{G}_1,\ldots,\mathcal{G}_t)$. In each turn, picks edge initially uncoloured $K_n$ Painter colours with colour $i \in \left\{ 1,\ldots,t \right\}$ her choice. The ends when a graph $G_i$ in $ for $G_i \mathcal{G}_i$ $i$ is crea...

Abstract Given coprime positive integers $$g_1< \ldots < g_e$$ g 1 < … e , the Frobenius number $$F=F(g_1,\ldots ,g_e)$$ F = ( ,...

By a compact packing of the plane by discs, $P$, we mean collection closed discs in with pairwise disjoint interior so that, for every disc $C\in P$, there exists sequence $D_{0},\ldots,D_{m-1}\in P$ that each $D_{i}$ is tangent to both $C$ and $D_{i+1\mod m}.$ We prove, $n\in\mathbb N$, exist only finitely many tuples $(r_{0},r_{1},\ldots,r_{n-1})\in\mathbb{R}^{n}$ $0<r_{0}<r_{1}\ldots<r_{n-1}...

A partition of the set $[n]:=\{1,2,\ldots,n\}$ is a collection disjoint nonempty subsets (or blocks) $[n]$, whose union $[n]$. In this paper we consider following rarely used representation for partitions: given $[n]$ with blocks $B_{1},B_{2},\ldots,B_{m}$ satisfying $\max B_{1}<\max B_{2}<\cdots<\max B_{m}$, represent it by word $w=w_{1}w_{2}\ldots w_{n}$ such that $i\in B_{w_{i}}$, $1\leq i\l...

Let $G= (V,E)$ be a $(p,q)$-graph. A bijection $f: Eto{1,2,3,ldots,q }$ is called an edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ where $f^+(u) = sum_{uwin E} f(uw)$. Moreover, a bijection $f: Eto{1,2,3,ldots,q }$ is called a semi-edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ or $f^+(u)=f^+(v)$. A graph that admits an&nbsp; ...

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets positive integers. $g_{\mathcal{B} }(n)$ count the number representations $n$ in form $n b_1\cdots b_h$, where $b_i \in B_i$ for all $i \{1,\ldots, h\}$. It is proved that $\liminf_{n\rightarrow \infty} g_{\mathcal{B} }(n) \geq 2$ implies $\limsup_{n\rightarrow \infty$.