平面图覆盖时猜想的数值证据
Numerical evidence against a conjecture on the cover time of planar graphs
摘要：
利用大型Monte Carlo模拟研究了平面图覆盖时间的一个猜想。猜想指出由$N$顶点和最大度$d$构成的平面图$G_{N}$的覆盖时间$τ(G_{N})$是由$τ(G_{N}) \geq C_{d} N(\ln N)^2$和$C_{d} = (d/4π) \tan (π/d)$构成的下界，等式对某些几何成立.我们在规则蜂窝（$d=3$）、规则正方形（$d=4$）、规则细长三角形（$d=5$）和规则三角形（$d=6$）晶格以及非规则米字格（$d_{\rm min}=4$、$d_{\rm max}=8$）上测试了这个猜想。事实上，蒙特卡罗数据表明，严格的下限可能会保持为大多数这些晶格的平等，在英国国旗晶格的情况下有一个有趣的问题。然而，蜂窝晶格的数据违反了约束常数的界限。还简要介绍了正方形格子的覆盖时间的经验概率分布函数，因为很少有人知道覆盖时间的概率分布函数一般。
英文摘要：
We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time $τ(G_{N})$ of a planar graph $G_{N}$ of $N$ vertices and maximal degree $d$ is lower bounded by $τ(G_{N}) \geq C_{d} N(\ln N)^2$ with $C_{d} = (d/4π) \tan (π/d)$, with equality holding for some geometries. We tested this conjecture on the regular honeycomb ($d=3$), regular square ($d=4$), regular elongated triangular ($d=5$), and regular triangular ($d=6$) lattices, as well as on the nonregular Union Jack lattice ($d_{\rm min}=4$, $d_{\rm max}=8$). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violates the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.
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