We study the sampling problem for simultaneous edge colorings. Given a pair of graphs $G_1=(V,E_1)$ and $G_2=(V,E_2)$ which are on the same vertex set $V$, a simultaneous edge coloring is an edge coloring of $G_1\cup G_2$ so that each of the individual graphs is properly colored. When each of $G_1$ and $G_2$ are of maximum degree $Δ$, then it is conjectured that $Δ+2$ colors suffice, and recent work asymptotically establishes the conjecture. We study Markov chains for randomly sampling from the uniform distribution over simultaneous edge colorings. Straightforward applications of Jerrum's classical coupling argument establish rapid mixing of the Glauber dynamics on the corresponding line graph when $k>8Δ$. We present a simple weighted Hamming distance for which Jerrum's coupling yields optimal mixing time (up to constant factors) of $O(m\log{n})$ when $k>(6+δ)Δ$ for any fixed $δ>0$. Moreover, utilizing the flip dynamics with our new metric, we obtain $O(m\log{n})$ mixing of the flip dynamics with a local choice of flip parameters, which flips only bounded-size components, when $k\geq 5.95Δ$. The proof adapts previous coupling analyses for the flip dynamics to the setting of simultaneous edge colorings.

翻译：我们研究同时边着色的采样问题。给定一对定义在相同顶点集$V$上的图$G_1=(V,E_1)$和$G_2=(V,E_2)$，同时边着色是对$G_1\cup G_2$的一种边着色，使得每个单独图均为正常着色。当$G_1$和$G_2$的最大度均为$\Delta$时，猜想$\Delta+2$种颜色足够，且近期工作已渐近地证实该猜想。我们研究用于从同时边着色的均匀分布中随机采样的马尔可夫链。直接应用Jerrum的经典耦合论证表明，当$k>8\Delta$时，对应线图上的Glauber动力学具有快速混合性。我们提出一种简单的加权汉明距离，在该距离下，对于任意固定$δ>0$，当$k>(6+δ)\Delta$时，Jerrum的耦合方法可获得$O(m\log{n})$的最优混合时间（至多常数因子）。此外，利用我们新度量下的翻转动力学，通过局部选择仅翻转有界大小分量的翻转参数，当$k\geq 5.95\Delta$时，可获得$O(m\log{n})$的混合时间。该证明将先前针对翻转动力学的耦合分析适配至同时边着色的设定。