Open Problem Garden

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational.

Conjecture There does not exist a quartic rationally derived polynomial $p \in {\mathbb Q}[x]$ with four distinct roots.

Keywords: derivative; diophantine; elliptic; polynomial

Posted by mdevos
updated February 16th, 2011

1 comment

Iradmusa, Moharram

Graph Theory » Coloring » Vertex coloring

Let $G$ be a simple graph, and for every list assignment $\mathcal{L}$ let $\lambda_{\mathcal{L}}$ be the maximum number of vertices of $G$ which are colorable with respect to $\mathcal{L}$ . Define $\lambda_t = \min{ \lambda_{\mathcal{L}} }$ , where the minimum is taken over all list assignments $\mathcal{L}$ with $|\mathcal{L}| = t$ for all $v \in V(G)$ .

Conjecture [2] Let $G$ be a graph with list chromatic number $\chi_\ell$ and $1\leq r\leq s\leq \chi_\ell$ . Then $\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.$