Abstract: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 when $k\geq 5.948\Delta$, using a local choice of flip parameters which only flips bounded-size components. The proof adapts previous coupling analyses for the flip dynamics to the setting of simultaneous edge colorings.
From: Ezra Furtado-Tiwari [view email]
[v1]
Wed, 6 May 2026 15:42:38 UTC (21 KB)
[v2]
Fri, 10 Jul 2026 14:02:38 UTC (22 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。