Tree-width and dimension

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph

Spectral gap for random-to-random shuffling on linear extensions

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of the Markov Chain as evidence + misc minor improvements; v4: fixed bibliograph