AI 中文总结
研究由分数色数和局部色数界定的图族的覆盖数问题，证明分数色数至多为$\beta$时无类似公式，给出上下界，还研究了特定图的覆盖数，得到局部色数至多为$k$时关于$\psi$的上界和关于$\omega$的下界。
AI 中文摘要
图类$\mathcal P$对图的覆盖数是覆盖其边所需的最少$\mathcal P$-图的数量。哈拉里、许和米勒的一个经典定理给出了色数至多为$k$的图类的覆盖数的精确公式。我们研究了分数色数$\chi_f$和局部色数$\psi$情况下的类似问题。我们证明了分数色数至多为$\beta$的图的覆盖数不存在类似公式，并找到了一个下界和一个上界，这引发了有趣的渐近问题。我们还研究了小的特定图的这种覆盖数。对于局部色数至多为$k$的图的覆盖数，我们找到了一个关于$\psi$的上界和一个关于$\omega$的下界。
英文摘要
The cover number of a graph by a graph class $\mathcal P$ is the least number of $\mathcal P$-graphs necessary to cover its edges. A classical theorem of Harary, Hsu and Miller gives an exact formula for the cover number by the class of graphs with chromatic number at most $k$. We investigate analogous questions for the case of the fractional chromatic number $χ_f$ and the local chromatic number $ψ$. We prove that an analogous formula cannot hold in the case of the cover number by graphs of fractional chromatic number at most $β$, and find a lower and an upper bound, that gives rise to interesting asymptotic questions. We also investigate this cover number for small specific graphs. In the case of the cover number by graphs with local chromatic number at most $k$, we find an upper bound in terms of $ψ$, and a lower bound in terms of $ω$.
