On Sendov's conjecture, M. J. Miller states the following in his paper [10,11]; if a zero $\beta$ of a polynomial which has all the zeros in the closed unit disk is sufficiently close to the unit circle, then the distance from $\beta$ to the closest critical point is less than or equal to 1. It is desirable to quantify this assertion. In this paper, we estimate the radius of the disk with center at 0 containing all the critical points and estimate the range of the zero $\beta$ satisfying the above for the first step. This result, moreover, implies that if Sendov’s conjecture is false, then the polynomial must be close to an extremal one.

A proper vertex coloring of a simple graph $G$ is $k$-forested if the subgraph induced by the vertices of any two color classes is a forest with maximum degree at most $k$. The $k$-forested chromatic number of a graph $G$, denoted by $\chi^{a}_{k}(G)$, is the smallest number of colors in a $k$-forested coloring of $G$. In this paper, it is shown that planar graphs with large enough girth do satisfy $\chi^{a}_{k}(G)=\lceil\frac{\Delta(G)}{k}\rceil+1$ for all $\Delta(G)> k\geq 2$, and $\chi^{a}_{k}(G)\leq 3$ for all $\Delta(G)\leq k$ with the bound 3 being sharp. Furthermore, a conjecture on $k$-frugal chromatic number raised in [1] has been partially confirmed.

Let $(F_{n})_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_{n}$, for $n\geq 0$, where $F_{0}=0$ and $F_{1}=1$. In this note, we prove that if $s$ is an integer number such that $F_{n}^{s}+F_{n+1}^{s}$ is a Fibonacci number for all sufficiently large integer $n$, then $s=1$ or 2.

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_{q}(\mathfrak{q}(n))$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for $U_{q}(\mathfrak{q}(n))$-modules in the category $\mathcal{O}_{\textit{int}}^{\ge 0}$.