Sampling Simultaneous Edge-Colorings

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 $\Delta$, then it is conjectured that $\Delta+2$ 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 $k>8\Delta$. 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 $k>(6+\delta)\Delta$ for any fixed $\delta>0$. 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 $k\geq 5.95\Delta$. The proof adapts previous coupling analyses for the flip dynamics to the setting of simultaneous edge colorings.