The Strong Fractional Choice Number and the Strong Fractional Paint Number of Graphs

This paper studies the strong fractional choice number $ch^s_f(G)$ and paint $pt^s_f(G)$ of a graph $G$. We prove that these parameters any finite are rational numbers. On other hand, for positive integers $p,q$ satisfying $2 \le \frac{2p}{2q+1} \leq \lfloor\frac{p}{q}\rfloor$, we construct $G$ with $ch^s_f(G) = pt^s_f(G) \frac{p}{q}$. The relationship between is explored. gap $pt^s_f(G)-ch^s_f(G)$ can be arbitrarily large. family $\mathcal{G}$ graphs supremum numbers in $\mathcal{G}$. Let $\mathcal{P}$ denote class planar $\mathcal{P}_{k_1,\ldots, k_q}$ without $k_i$-cycles $i=1,\ldots, q$. $3 + \frac{1}{2} ch^s_f(\mathcal{P}_{ 4}) 4$, $ch^s_f(\mathcal{P}_{ k})=4$ $k \in \{5,6\}$, +\frac{1}{12} 4,5}) $ch^s_f(\mathcal{P}) \ge 4+\frac 13$. last result improves lower bound $4+\frac 29$ [Zhu, J. Combin. Theory Ser. B, 122 (2017), pp. 794--799].